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Вестник Челябинского государственного университета. 2015. № 3 (358). Математика. Механика. Информатика. Вып. 17. С. 67-117.
УДК 515.163 ББК В151.5
AN INTRODUCTION TO FINITE TYPE INVARIANTS OF KNOTS AND 3-MANIFOLDS DEFINED BY COUNTING GRAPH CONFIGURATIONS
C. Lescop
The finite type invariant concept for knots was introduced in the 90's in order to classify knot invariants, with the work of Vassiliev, Goussarov and Bar-Natan, shortly after the birth of numerous quantum knot invariants. This very useful concept was extended to 3-manifold invariants by Ohtsuki.
These introductory lectures show how to define finite type invariants of links and 3-manifolds by counting graph configurations in 3-manifolds, following ideas of Witten and Kontsevich.
The linking number is the simplest finite type invariant for 2-component links. It is defined in many equivalent ways in the first section. As an important example, we present it as the algebraic intersection of a torus and a 4-chain called a propagator in a configuration space.
In the second section, we introduce the simplest finite type 3-manifold invariant, which is the Casson invariant (or the ©-invariant) of integer homology 3-spheres. It is defined as the algebraic intersection of three propagators in the same two-point configuration space.
In the third section, we explain the general notion of finite type invariants and introduce relevant spaces of Feynman Jacobi diagrams.
In Sections 4 and 5, we sketch an original construction based on configuration space integrals of universal finite type invariants for links in rational homology 3-spheres and we state open problems. Our construction generalizes the known constructions for links in M3 and for rational homology 3-spheres, and it makes them more flexible.
In Section 6, we present the needed properties of parallelizations of 3-manifolds and associated Pontrjagin classes, in details.
Keywords: knots, 3-manifolds, finite type invariants, homology 3-spheres, linking number, Theta invariant,Casson-Walker invariant,Feynman Jacobi diagrams,perturbative expansion of Chern— Simons theory,configuration space integrals,parallelizations of 3-manifolds,first Pontrjagin class.
1. Various aspects of the linking number......................................................................68
1.1. The Gauss linking number of two disjoint knots in R3 the ambient space.................68
1.2. Some background material on manifolds without boundary, orientations, and degree.... 69
1.3. The Gauss linking number as a degree................................................................70
1.4. Some background material on manifolds with boundary and algebraic intersections ...71
1.5. A general definition of the linking number..........................................................72
1.6. Generalizing the Gauss definition of the linking number
and identifying the definitions........................................................................... 72
2. Propagators and the ©-invariant..............................................................................74
2.1. Blowing up in real differential topology.............................................................74
2.2. The configuration space C2(M)..........................................................................74
2.3. On propagators...............................................................................................77
2.4. The ©-invariant of (M, t).................................................................................77
2.5. Parallelisations of 3-manifolds and Pontrjagin classes...........................................78
2.6. Defining a Q-sphere invariant from ©.................................................................79
3. An introduction to finite type invariants................................................................... 81
3.1. Lagrangian-preserving surgeries.........................................................................81
3.2. Definition of finite type invariants.....................................................................82
3.3. Introduction to chord diagrams.........................................................................84
3.4. More spaces of diagrams...................................................................................85
3.5. Multiplying diagrams.......................................................................................87
4. Configuration space construction of universal finite type invariants..............................89
4.1. Configuration spaces of links in 3-manifolds........................................................89
4.2. Configuration space integrals............................................................................89
4.3. An invariant for links in Q-spheres from configuration spaces ................................91
4.4. On the universality proofs................................................................................92
5. Compactifications, anomalies, proofs and questions....................................................93
5.1. Compactifications of configuration spaces...........................................................94
5.2. Straight links.................................................................................................95
5.3. Rationality of Z..............................................................................................96
5.4. On the anomalies ............................................................................................ 96
5.5. The dependence on the forms in the invariance proofs...........................................97
5.6. The dependence on the parallelizations in the invariance proofs............................ 100
5.7. End of the proof of Theorem 5........................................................................ 101
5.8. Some open questions...................................................................................... 102
6. More on parallelizations of 3-manifolds and Pontrjagin classes................................... 103
6.1. [(M,dM),(SO(3),1)] is an abelian group......................................................... 103
6.2. Any oriented 3-manifold is parallelizable.......................................................... 104
6.3. The homomorphism induced by the degree on [(M, dM),(SO(3),1)]..................... 105
6.4. First homotopy groups of the groups SU(n)...................................................... 106
6.5. Definition of relative Pontrjagin numbers......................................................... 107
6.6. On the groups SO(3) and SO(4)..................................................................... 108
6.7. Relating the relative Pontrjagin number to the degree........................................ 110
7. Other complements ............................................................................................. 111
7.1.More on low-dimensional manifolds.................................................................. 112
Foreword
These notes contain some details about talks that were presented in the international conference "Quantum Topology" organized by Laboratory of Quantum Topology of Chelyabinsk State University in July 2014. They are based on the notes of five lectures presented in the ICPAM-ICTP research school of Meknes in May 2012. I thank the organizers of these two great events. I also thank Catherine Gille and Kevin Corbineau for useful comments on these notes.
These notes have been written in an introductory way, in order to be understandable by graduate students. In particular, Sections 1, 2 and 6 provide an elementary self-contained presentation of the ©-invariant. The notes also contain original statements (Theorems 5, 6, 7 and 8) together with sketches of proofs. Complete proofs of these statements, which generalize known statements, will be included in a monograph [1].
1. Various aspects of the linking number
1.1. The Gauss linking number of two disjoint knots in R3 the ambient space
The modern powerful invariants of links and 3-manifolds that will be defined in Section 4 can be thought of as generalizations of the linking number. In this section, we warm up with several ways of defining this classical basic invariant. This allows us to introduce conventions and methods that will be useful througout the article.
Let S1 denote the unit circle of C: S = {z;z e C,|z| = 1}. Consider two C™ embeddings J: S1 - R3 and K: S1 — R3 \ /(S1)
\
>K I " /
and the associated Gauss map
pJK : S1 x S1 S2,
(w, z) ^
1
K(z) - J(w).
(K(z) - J(w))
Pjk
S2 so that ®S2 is the homogeneous volume
Denote the standard area form of S2 by 4n® form of S2 such that J 2 = 1. In 1833, Gauss defined the linking number of the disjoint
knots J(S4) and K(S4), simply denoted by J and K, as an integral [2]. With modern notation, his definition reads
lkG (J, K) = jSixSiPji< (®s2>.
It can be rephrased as lkG(j, K) is the degree of the Gauss map pJK.
1.2. Some background material on manifolds without boundary, orientations, and degree
A topological n-dimensional manifold M without boundary is a Hausdorff topological space that is a countable union of open subsets Ut labeled in a set I (i e I), where every Ut is identified with an open subset Vi of K" by a homeomorphism : Ut ^ V, called a chart. Manifolds are considered up to homeomorphism so that homeomorphic manifolds are considered identical.
For r = 0,...,ro, the topological manifold M has a Cr-structure or is a Cr-manifold, if, for each pair {i,j} c I, the map ^ ° defined on ^t(Ut o Uy) is a Cr-diffeomorphism to its image. The notion of Cs-maps, s < r, from such a manifold to another one can be naturally deduced from the known case where the manifolds are open subsets of some K", thanks to the local identifications provided by the charts. Cr-manifolds are considered up to Cr-diffeomorphisms.
An orientation of a real vector space V of positive dimension is a basis of V up to a change of basis with positive determinant. When V = {0}, an orientation of V is an element of {-1,1}. For " > 0, an orientation of K" identifies Hn(Kn,Kn \{x};K) with K. (In these notes, we freely use basic algebraic topology, see [3] for example.) A homeomorphism h from an open subset U of K" to another such V is orientation-preserving at a point x, if h*: Hn (U,U \{x}) ^ Hn (V,V \{h(x)}) is orientation-preserving. If h is a diffeomorphism, h is orientation-preserving at x if and only if the determinant of the Jacobian Txh is positive. If K" is oriented and if the transition maps ^j ° are orientation-preserving (at every point) for {i,j} c I, the manifold M is oriented.
For " = 0, 1, 2 or 3, any topological "-manifold may be equipped with a unique smooth structure (up to diffeomorphism) (See Theorem 10, below). Unless otherwise mentioned, our manifolds are smooth (i. e. Cw), oriented and compact, and considered up oriented diffeomor-phisms. Products are oriented by the order of the factors. More generally, unless otherwise mentioned, the order of appearance of coordinates or parameters orients manifolds.
J
A point y is a regular value of a smooth map p: M ^ N between two smooth manifolds M and N, if for any x e p l(y) the tangent map Txp at x is surjective. According to the Morse — Sard theorem [4. P. 69], the set of regular values of such a map is dense. If M is compact, it is furthermore open.
When M is oriented and compact, and when the dimension of M coincides with the dimension of N, the differential degree of p at a regular value y of N is the (finite) sum running over the x e p l(y) of the signs of the determinants of Txp. In our case where M has no boundary, this differential degree is locally constant on the set of regular values, and it is the degree of p, if N is connected. See [5. Chapter 5].
Finally, recall a homological definition of the degree. Let [M] denote the class of an oriented closed (i. e. compact, connected, without boundary) n-manifold in Hn(M;Z). Hn(M;Z) = Z[M]. If M and N are two closed oriented n-manifolds and if f: M^N is a (continuous) map, then Hn(f)([M]) = deg(f)[N]. In particular, for the Gauss map pJK of Subsection 1.1, H2(pjk)([51x51]) = lk(J,K)[S2].
1.3. The Gauss linking number as a degree
Since the differential degree of the Gauss map pJK is locally constant, lkG(J,K) = [ 1 pJK(o>)
p J S xS
for any 2-form ® on S such that = 1.
Let us compute lkG(j,K) as the differential degree of pJK at the vector Y that points towards us. The set p-K(Y) is made of the pairs of points (w, z) where the projections of J(w) and K(z) coincide, and J(w) is under K(z). They correspond to the crossings J5K and K|J of the diagram.
In a diagram, a crossing is positive if we turn counterclockwise from the arrow at the end of the upper strand to the arrow of the end of the lower strand like 5. Otherwise, it is negative like | .
For the positive crossing J5K, moving J(w) along J following the orientation of J, moves pJK(w,z) towards the South-East direction, while moving K(z) along K following the orientation of K, moves pJK(w,z) towards the North-East direction, so that the local orientation
/*Tp dz y
induced by the image of pJK around YeS2 is d which is ^ . Therefore, the contribution of
Tp— *
dw
a positive crossing to the degree is 1. Similarly, the contribution of a negative crossing is (-1).
We have just proved the following formula
degY(pjK) = #J^K - #K|J where # stands for the cardinality — here # J^K is the number of occurences of J^K in the diagram — so that
lkG(j,K) = #J^K - #K|J.
Similarly, deg-Y(pJK) = #K|J - #J^K so that
lkG(j,K) = #K^J - #J^K = ^(#JKK + #K^J) - ^(#KIJ + #J^K) and lkG(J,K) = lkG(K,J).
In our first example, lkG(j,K) = 2. Let us draw some further examples.
For the positive H op f link lkG(j,K) =
For the negative H op f link lkG(j,K) =
For the Whitehead link ,lkr(j,K) = 0.
1.4. Some background material on manifolds with boundary and algebraic intersections
A topological n-dimensional manifold M with possible boundary is a Hausdorff topological space that is a union of open subsets Ui labeled in a set I, (i e I), where every Ui is identified with an open subset Vi of ]-ro,0]kxRn-k by a chart : U ^ Vi. The boundary of ]-ro,0]kxRn-k is made of the points (xp...,xn) of ]-ro,0]kxRn-k such that there exists i < k such that xt = 0. The boundary of M is made of the points that are mapped to the boundary of ]-ro,0]kxRn-k.
For r = 1,..., ro, the topological manifold M is a Cr-manifold with ridges (or with corners) (resp. with boundary), if, for each pair {i, j} c I, the map ^j ° ^r1 defined on ^.(U. n Uj) is a Cr-diffeomorphism to its image (resp. and if furthermore k < 1, for any i). Then the ridges of M are made of the points that are mapped to points (x1,...,xn) of ]-ro,0]kxRn-k so that there are at least two i < k such that xi = 0.
The tangent bundle to an oriented submanifold A in a manifold M at a point x is denoted by TxA. The normal bundle TxM / TxA to A in M at x is denoted by VxA. It is oriented so that (a lift of an oriented basis of) VxA followed by (an oriented basis of) TxA induce the orientation of TxM. The boundary dM of an oriented manifold M is oriented by the outward normal first convention. If x edM is not in a ridge, the outward normal to M at x followed by an oriented basis of TxdM induce the orientation of M. For example, the standard orientation of the disk in the plane induces the standard orientation of the circle, counterclockwise, as the following picture shows.
As another example, the sphere S2 is oriented as the boundary of the ball B3, which has the standard orientation induced by (Thumb, index finger (2), middle finger (3)) of the right hand.
Two submanifolds A and B in a manifold M are transverse if at each intersection point x, TxM = TxA + TxB. The transverse intersection of two submanifolds A and B in a manifold M is oriented so that the normal bundle to A n B is (V(A) © V(B)), fiberwise. If the two manifolds are of complementary dimensions, then the sign of an intersection point is +1 if the orientation of its normal bundle coincides with the orientation of the ambient space, that is if TxM = VxA © VxB (as oriented vector spaces), this is equivalent to TxM = TxA © TxB (as oriented vector spaces again, exercise). Otherwise, the sign is -1. If A and B are compact and if A and B are of complementary dimensions in M, their algebraic intersection is the sum of the signs of the intersection points, it is denoted by (A,B)M.
When M is an oriented manifold, (-M) denotes the same manifold, equipped with the opposite orientation. In a manifold M, a k-dimensional chain (resp. rational chain) is a finite combination with coefficients in Z (resp. in Q) of smooth k-dimensional oriented submanifolds C of M with boundary and ridges, up to the identification of (-1)C with (-C).
Again, unless otherwise mentioned, manifold are oriented. The boundary d of chains is a linear map that maps a smooth submanifold to its oriented boundary. The canonical orientation of a point is the sign +1 so that d[0,1] = {1} - {0}.
Lemma 1. Let A and B be two transverse submanifolds of a d-dimensional manifold M, of respective dimensions a and p, with disjoint boundaries. Then
d(A nB) = (-1)d-p dA nB + A ndB.
Proof. Note that d(A n B) c dA u dB. At a point a e dA, TaM is oriented by (VaA, o,TadA), where o is the outward normal to A. If a e dAnB, then o is also an outward normal to A nB, and d(A n B) is cooriented by (VaA, VaB, o) while dA n B is cooriented by (VaA, o, VaB). At a point b e A n dB, d(A n B) is cooriented by (VaA, VaB,o) like A n dB. □
1.5. A general definition of the linking number
Lemma 2. Let J and K be two rationally null-homologous disjoint cycles of respective dimensions j and k in a d-manifold M, where d = j + k + 1. There exists a rational (j + 1)-chain ZJ bounded by J transverse to K,and a rational (k + 1)-chain ZK bounded by K transverse to J and for any two such rational chains ZJ and ZK, (J, ZK )M = (-1)J+1 (ZJ,K)M. In particular, <J, ZK )M is a topological invariant of (J,K),which is denoted by lk(j,K) and called the linking number of J and K.
lk(JK) = (-i)(j+1)(k+1)lk(K,J).
Proof. Since K is rationally null-homologous, K bounds a rational (k + 1)-chain ZK. Without loss, zk is assumed to be transverse to Z^ so that Z^ nZ^ is a rational 1-chain (which is a rational combination of circles and intervals). (As explained in [4. Chapter 3], generically, manifolds are transverse). According to Lemma 1,
d(Zj n Zx ) = (-1)d+A+1 J n Zx + Zj n K.
Furthermore, the sum of the coefficients of the points in the left-hand side must be zero, since this sum vanishes for the boundary of an interval. This shows that (J,ZK)M = (-1) + (ZJ,K>M , and therefore that this rational number is independent of the chosen Zj and Z^. Since
(-DM(zj,K)M = (-iy+1(-i)«j+1>{k,zj)M, ik(j,k) = (-i)<^+1><k+1>ik(K, j). □
In particular, the linking number of two rationally null-homologous disjoint links J and K in a 3-manifold M is the algebraic intersection of a rational chain bounded by one of the knots and the other one.
For K = Z or Q, a "K-sphere or (integer or rational) homology 3-sphere (resp. a K-ball) is a smooth, compact, oriented 3-manifold, without ridges, with the same K-homology as the sphere 53 (resp. as a point). In such a manifold, any knot is rationally null-homologous so that the linking number of two disjoint knots always makes sense.
A meridian of a knot K is the (oriented) boundary of a disk that nK intersects K once with a positive sign. Since a chain Z^ bounded by a knot J disjoint from K in a 3-manifold M provides a rational cobordism between J and a combination of meridians of K, [/] = lk(j,K)[mK] in H] (M \ K:Q) where mK is a meridian of K.
Lemma 3. When K is a knot in a Q-sphere or a Q-ball M, H1 (M \K; Q) = Q[mK ], so that the equation [J]=lk(J,K)[mK] in H1 (M \K;Q) provides an alternative definition for the linking number.
Proof. Exercise. □
The reader is also invited to check that lkG = lk as an exercise though it will be proved in the next subsection, see Proposition 1.
1.6. Generalizing the Gauss definition of the linking number and identifying the definitions
Lemma 4. The map
p.: ((M3)2 \ diag) ^ S2
1
(x, y) ^ -(y - x)
II y - x II
is a homotopy equivalence. In particular H(pS2): H;((R3)2 \ diag;Z) ^ Hi(S'2;Z) is an isomorphism for all i, (R3)2 \ diag is a homology S2, and [5*] = (H2(PS2)) [52] is a canonical generator of H2((R3)2 \ diag;Z) = Z[S]. 5
Proof. The map (x,y) h-> (x, || y - x ||,p 2(x,y)) provides a homeomorphism from (R3)2 \ diag to R3x]0,<x[xS2. 5 □
As in Subsection 1.1, consider a two-component link J U K: S1 U S1 ^ R3. This embedding induces an embedding
J x K : S1 x S1 ^ ((R3)2 \ diag)
(zv z2) ^ (J(zf), K(z2))
the map pJK of Subsection 1.1 reads pS2°(/*K), and since H2(p/K)[S1xS1] = deg(p/K)[S2] = lkG(J,K)[S2] in H2(S2;Z) = Z[S2], [J x K] = H2(J x K)[Sl x S1] = lkG(J,K)[S] in H2((R3)2\diag;Z = (Z[S]. We will see that this definition of lkG generalizes to links in rational homology 3-spheres and then prove that our generalized definition coincides with the general definition of linking numbers in this case.
For a 3-manifold M, the normal bundle to the diagonal of M2 in M2 is identified with the tangent bundle to M, fiberwise, by the map
(uv) e di^oS?) "(v- u) e T'M-
A parallelization t of an oriented 3-manifold M is a bundle isomorphism t : M x R3 ^ TM that restricts to x x R3 as an orientation-preserving linear isomorphism from x x R3 to TxM, for any x e M. It has long been known that any oriented 3-manifold is parallelizable (i. e. admits a parallelization). (It is proved in Subsection 6.2.) Therefore, a tubular neighborhood of the diagonal in M2 is diffeomorphic to M x R3.
Lemma 5. Let M be a rational homology 3-sphere, let <x> be a point of M. Let M = (M \ {<»}). Then M2 \ diag has the same rational homology as S2. Let B be a ball in M and let x be a point inside B, then the class [S] of xxdB is a canonical generator of H2(M2 \diag Q) = Q[£].
Proof. In this proof, the homology coefficients are in Q. Since M has the homology of a point, the Kunneth Formula implies that M2 has the homology of a point. Now, by excision,
H(M2,MM2 \ diag) ^ H(M x R3,M x (R3 \ 0)) = H,(R3,S2) = JQ if * = 3,
[0 otherwise.
Using the long exact sequence of the pair (M2,M2 \diag, we get that H*(M2 \ diag;Q) = H„(S2). □
Define the Gauss linking number of two disjoint links J and K in M so that
[(J x K)(S1 x S1)] = lkG (J,K)[S] in H2(M2 \diag;Q. Note that the two definitions of lkG coincide when M = R3.
Proposition 1. lkG = lk.
Proof. First note that both definitions make sense when J and K are disjoint links: [.JxK] = lkG(Jand lk(j,K) is the algebraic intersection of K and a rational chain 2j bounded by J.
If K is a knot, then the chain 2j of M provides a rational cobordism C between J and a combination of meridians of K in M \K, and a rational cobordism CxK in M2\diag, which allow us to see that lkG(-,K) and lk(.,K) linearly depend on [J] e HX(M \K). Thus we are left with the proof that lkG(mK,K) = lk(mK,K) = 1. Since lkG(mK,.) linearly depends on [K] e HX(M \ mK), we are left with the proof lkG(mK,K) = 1 when K is a meridian of mK. Now, there is no loss in assuming that our link is a Hopf link in R3 so that the equality follows from the equality for the positive Hopf link in R3. □
For a 2-component link (J,K) in R3, the definition of lk(j,K) can be rewritten as
lk(j, k) = l/s ^ = <j x pi(yk 3)2xdiag
for any regular value Y of pJK, and for any 2-form ® of S2 such that J2® = 1. Thus, lk(J,K)
is the evaluation of a 2-form p*2(®) of (R3)2 \ diag at the 2-cycle [JXK], or it is the intersection of the 2-cycle [JXK] with a 4-manifold p-2 (Y), which will later be seen as the interior of a prototypical propagator. We will adapt these definitions to rational homology 3-spheres in Subsection 2.3. The definition of the linking number that we will generalize in order to produce more powerful invariants is contained in Lemma 8.
2. Propagators and the ©-invariant
Propagators will be the key ingredient to define powerful invariants from graph configurations in Section 4. They are defined in Subsection 2.3 below after needed preliminaries. They allow us to define the ©-invariant as an invariant of parallelized homology 3-balls in Subsection 2.4. The ©-invariant is next turned to an invariant of rational homology 3-spheres in Subsection 2.6.
2.1. Blowing up in real differential topology
Let A be a submanifold of a smooth manifold B, and let UV(A) denote its unit normal bundle. The fiber UVa(A) = (Va(A)\{0})/R+* of UV(A) is oriented as the boundary of a unit ball of V (A).
Here, blowing up such a submanifold A of codimension c of B means replacing A by UV(A). For small open subspaces UA of A, ((Rc = {0}u]0,<»[xSc-1)xUA) is replaced by ([0,«[xSc-1 xUA), so that the blown-up manifold Bl(B,A) is homeomorphic to the complement in B of an open tubular neighborhood (thought of as infinitely small) of A. In particular, Bi(B,A) is homoto-py equivalent to B \ A. Furthermore, the blow up is canonical, so that the created boundary is ±UV(A) and there is a canonical smooth projection from Bl(B,A) to B such that the preimage of ae A is UVa(A). If A and B are compact, then Bi(B,A) is compact, it is a smooth compac-tification of B \ A.
In the following figure, we see the result of blowing up (0,0) in R2, and the closures in ^(M2,(0,0)) of {0}xR, Rx{0} and the diagonal of R2, successively.
2.2. The configuration space C2(M)
See S3 as R3 u<x> or as two copies of R3 identified along R3 \{0} by the (exceptionally orientation-reversing) diffeomorphism x ^ x/1| x ||2. Then B£(S3,co) = R3 u S2 where the unit normal bundle (-S2) to ro in S3 is canonically diffeomorphic to S2 via p2 : ^ S2, where x e S^ is the limit of a sequence of points of R3 approaching ro along a line directed by p2 (x) e S2, 6B£(S V) = Si
Fix a rational homology 3-sphere M, a point ro of M, and M = M \{<»}. Identify a neighborhood of ro in M with the complement B12 of the closed ball B(1) of radius 1 in R3. Let B22 be the complement of the closed ball B(2) of radius 2 in R3, which is a smaller neighborhood of
ro in M via the understood identification. Then BM = M\B2w is a compact rational homology ball diffeomorphic to B£(M, ro).
Define the configuration space C2(M) as the compact 6-manifold with boundary and ridges obtained from M2 by blowing up (ro,ro), the closures in B£(M2,(ro,ro)) of {ro} xM, M x {ro} and the diagonal of M2 successively. Then dC2(M) contains the unit normal bundle
TM2 diag
\ {0}
/
to the diagonal of M2. This bundle is canonically isomorphic to the unit tangent bundle UM to M (again via the map ([(x, y)] ^ [y - x])).
Lemma 6. Let C2(M) = M2\ diag. The open manifold C2(M)\dC2(M) is C2(M) and the inclusion C2(M) ^C2(M) is a homotopy equivalence. In particular,C2(M) is a compactifi-cation of C2(M) homotopy equivalent to C2(M). The manifold C2(M) is a smooth compact 6-dimensional manifold with boundary and ridges. There is a canonical smooth projection Pm2 : C2(M) ^M2, dC2(M) = (Si xM) u (-M x Si) u UM ± p^2 (ro,ro).
Proof. Let B1ro be the complement of the open ball of radius one of M3 in S3. Blowing up (ro, ro) in 5j2ro transforms a neighborhood of (ro, ro) into the product [0,1[x^5. Explicitly, there is a map y :[0,1[x5'5 ^ b£(B2„ ,(ro, ro)), where B£(B2„ ,(ro, ro)) c B£(M2 ,(ro, ro)) ,suchthat when Xe]0,1[ and (x,y) is an element of the unit sphere S5 of (M3)2 such that x ^ 0 and y ± 0,
( i i
y(X,(x, y)) =
y
vX || x || X || y n
and such that y is a diffeomorphism onto its image, which is a neighborhood of the preima-
ge of (ro, ro) under the blow-down map B£(M2,(ro,ro)) ^^ M2. This neighborhood intersects roxM, M xro, and diag(M2) as y(]0,1[x0 x S2), y(]0,1[xS2 x 0) and y(]0,1[x(S5 n diag ((M3)2))), respectively. In particular, the closures of roxM, M xro, and diag (M2) in in-
tersect the boundary y(0 x S ) of B£(M ,(ro, ro)) as three disjoint spheres in S5, and they read rox B£(M, ro), B£(M, ro) xro and diag (B£(M, ro)2). Thus, the next steps will be three blow-ups along these three disjoint smooth manifolds.
These blow-ups will preserve the product structure y([0,1[x-). Therefore, C2(M) is a smooth compact 6-dimensional manifold with boundary, with three ridges S x S in P-2(ro,ro). A neighborhood of these ridges in C2(M) is diffeomorphic to [0,1[2 xS2 x S2. □
Lemma 7. The map ps2 of Lemma 4 smoothly extends to C2(S3), and its extension p^ satisfies:
-pm o Pi on S» X M3, PS2 = jp*> o P2 on M3 X S2,
[p2 on UM3 = M3 X S2,
where pl and p2 denote the projections on the first and second factor with respect to the above expressions.
Proof. Near the diagonal of M3, we have a chart of C2(S3)
yd : M3 x [0,ro[xS2 ^ C2(S3) that maps (x eM3,Xe]0,ro[,y eS2) to (x,x + Xj) e (M3)2. Here, p 2 extends as the projection onto the S2 factor.
Consider the orientation-reversing embedding
A :
Too
|(x e S2) ^
S3,
co
if | = 0,
— x otherwise. I
Note that this chart induces the already mentioned identification px of the ill-oriented unit normal bundle S2x to M in S3 with S2. When | ^ 0,
p,(4>„ (mx), y e M3) = My - .
S || My - x ||
Then p2 can be smoothly extended on S^ xM3 (where | = 0) by p 2(x e S2,y e M3) = -x. Similarly, set Ps2(x e M3,y e S2) = y. Now, with the map y of the proof of Lemma 6, when x and y are not equal to zero and when they are distinct,
y _ x
p ° (x,y)))=i ^ii=iii x i2 y-II yi2 x i
II y i2 II x i2
when A ^ 0. This map naturally extends to B£(M2,(ro, ro)) outside the boundaries of rox Bt(M ,ro), Bt(M, ro) xro and diag (Bl(M, ro)) by keeping the same formula when A = 0.
Let us check that p 2 smoothly extends over the boundary of the diagonal of Bl(M,ro). There is a chart of C2(M) near the preimage of this boundary in C2(M)
y2 : [0,ro[x[0,ro[xS2 x S2 ^ C2(S3) that maps (Ae]0, ro[, |e]0, ro[, x e S2, y e S2) to (Ax), (A(x + |>0)) where p2 reads
x y - 2(x, y)x - ux
(A, u, x, y) ^ —-,
|| y - 2{x, y)x - ux ||
and therefore smoothly extends when | = 0. We similarly check that p 2 smoothly extends over the boundaries of (rox Bl(M, ro)) and (Bt(M ,ro) xro). □
Let tj denote the standard parallelization of M3. Say that a parallelization
t : M x M3 ^ TM
of M that coincides with ts on B1ro is asymptotically standard. According to Subsection 6.2, such a parallelization exists. Such a parallelization identifies UM with M x S .
Proposition 2. For any asymptotically standard parallelization t of M, there exists a smooth map pT: dC2(M) ^ S2 such that
Pt
PS2 on Pm^'
- Pi ° Pi on Si x M, Pi ° P2 0n M X Sl'
p2 on UM = M X S2,
where p1 and p2 denote the projections on the first and second factor with respect to the above expressions.
Proof. This is a consequence of Lemma 7. □
Since C2(M) is homotopy equivalent to (M2\ diag), according to Lemma 5, H2(C2(M); Q) = Q[^] where the canonical generator [S] is the homology class of a fiber of UM cdC2(M). For a 2-component link (J,K) of M, the homology class [J xK] of J xK in H2(C2(M); Q) reads lk(j,K)[S], according to Subsection 1.6 and to Proposition 1.
Define an asymptotic rational homology M3 as a pair (M, t) where M is 3-manifold that reads as the union over ]1,2] x S2 of a rational homology ball BM and the complement B1ro of the unit ball of M3, and t is an asymptotically standard parallelization of M. Since such a pair (M,t) canonically defines the rational homology 3 -sphere M = M u {ro}, "Let (M,t) be an asymptotic rational homology M3" is a shortcut for "Let M be a rational homology 3-sphere equipped with an asymptotically standard parallelization t of M".
2.3. On propagators
Definition 1. Let (M, t) be an asymptotic rational homology M3. A propagating chain of (C2(M),t) is a 4-chain V of C2(M) such that dV = p-\a) for some a e S2. A propagating form of (C2(M),t) is a closed 2-form rop on C2(M) whose restriction to dC2(M) reads p*(ro) for some 2-form ro of S2 such that j2ro = 1. Propagating chains and propagating forms are simply called propagators when their nature is clear from the context.
Example 1. Recall the map p 2 : C2(S3) ^S2 of Lemma 7. For any a e S2, p-2(a) is a propagating chain of (C2(S3),ts), and for any 2-form ro of S2 such that j 2ro = 1, p*2(ro) is a propagating form of (C2(S3), ts ).
Propagating chains exist because the 3-cycle p-1(a) of dC2(M) bounds in C2(M) since H3(C2(M);Q) = 0. Dually, propagating forms exist because the restriction induces a surjective map H2(C2(M);M) ^ H2(dC2(M);M) since H3(C2(M),dC2(M);M) = 0 . Explicit constructions of propagating chains associated to Morse functions or Heegaard diagrams can be found in [6].
Lemma 8. Let (M,t) be an asymptotic rational homology M3. Let C be a two-cycle of C2(M). For any propagating chain V of (C2(M),t) transverse to C and for any propagating
form rop of (C2(M),t), [c] = jrop[s] = <c,V)^(M)[s] in H2(C2(M);Q) = Q[5]. In particular,for
any two-component link (J,K) of M lk(J,K) = j ^ro p =<J x K,V)C (M
Proof. Fix a propagating chain V, the algebraic intersection <C, V)C (M) only depends on the homology class [C] of C in C2(M). Similarly, since rop is closed, jrop only depends on [C]. (Indeed, if C and C cobound a chain D, Cn V and C n V cobound ±(Dn V), and i®-c rop = jdrop according to the Stokes theorem.) Furthermore, the dependance on [C] is
linear. Therefore it suffices to check the lemma for a cycle that represents the canonical generator [S] of H2(C2(M);Q). Any fiber of UM is such a cycle. □
2.4. The ©-invariant of (M,t)
Note that the intersection of transverse (oriented) submanifolds is an associative operation, so that A n B n C is well defined. Furthermore, for a connected manifold N, the class of a 0-cycle in H0(M;Q) = Q[/m] = Q is a well-defined number, so that the algebraic intersection of several transverse submanifolds whose codimension sum is the dimension of the ambient manifold is well defined as the homology class of their (oriented) intersection. This extends to rational chains, multilinearly. Thus, for three such transverse submanifolds A, B, C in a manifold D, their algebraic intersection <A,B, C)D is the sum over the intersection points a of the associated signs, where the sign of a is positive if and only if the orientation of D is induced by the orientation of VaA © VaB © VaC .
Theorem 1. Let (M,t) be an asymptotic rational homology M3. Let Va> Vb and Vc be three pairwise transverse propagators of (C2(M),t) with respective boundaries pTll(a), p-1 (b) and pTl1(c) for three distinct points a,b and c of S2,then ©(M,t) = <Va,Vb,VC)c (M) does not depend on the chosen propagators Va, Vb and Vc. It is a topological invariant of (M,t). For any three propagating chains roa, rob and roc of (C2(M),t),
©(M,t) = i ro A ro, A ro .
v ' ' Jc2(m) a b c
Proof. Since H4(C2(M)) = 0, if the propagator Va is changed to a propagator Pa with the same boundary, (V - V) bounds a 5-dimensional chain W transverse to Vb n V.. The 1-dimensional chain Wn Vb n Vc does not meet dC2(M) since Vb n Vc does not meet dC2(M). Therefore, up to a well-determined sign, the boundary of W n Vb n Vc is Va n Vb n Vc - Va n Vb n Vc. This shows that <Va, Vb, Vc)c (M) is independent of Va when a is fixed. Similarly, it is independent of Vb and Vc when b and c are fixed. Thus, <Va, Vb, Vc)C (M) is a rational function on the connected set of triples (a, b,c) of distinct point of S2. It is easy to see that this function is continuous. Thus, it is constant.
Let us similarly prove that jc (M)®a Aab A®c is independent of the propagating forms ®a, ®b
and ®c. Assume that the form ®a, which restricts to dC2(M) as p* ), is changed to <a'a, which restricts to dC2(M) as p*(<'A).
Lemma 9. There exists a one-form nA on S2 such that <A = <A + dnA. For any such nA, there exists a one-form n on C2(M) such that <a-<a = dn, and the restriction of n to dC2(M) is p*(nA).
Proof. Since ®a and <'a are cohomologous, there exists a one-form n on C2(M) such that <a = <a + dn. Similarly, since j 2a>'A = j 2roA, there exists a one-form nA on S2 such that <A = <A + dnA. On dC2(M), d(n-p*(n)) = 0. Thanks to the exact sequence
0 = H1 (C2 (M)) ^ H1 (dC2 (M)) ^ H2(C2 (M), 3C2 (M)) = H4 (C2 (M)) = 0, Hl(dC2(M)) = 0. Therefore, there exists a function ffrom dC2(M) to R such that df = n-p(T)"(nA) on dC2(M). Extend f to a Cm map on C2(M) and change n into (n-df). □
Then
I < A<A< -I < A<A< =1 d(tlA<Affl ) =
JC2(M) a b c JC2(M) a b c JC2(M) v 1 b
= I nA<Affl =I p(T)* (nA<A< ) = 0
J3C2(M)l b c Jac2(mtA B c>
since any 5-form on S2 vanishes. Thus, jc (M<a A<b a®c is independent of the propagating
forms ®a, <b and ®c. Now, we can choose the propagating forms ®a, <b and ®c, Poincare dual to Va, Vb and Vc, and supported in very small neighborhoods of Va, Vb and Vc, respectively, so that the intersection of the three supports is a very small neighborhood of Va n Vb n Vc, where it can easily be seen that j m< A<b A®c = (Va,Vb,Vc)c2(M). □
In particular, ©(M,t) reads jc (M)®3 for any propagating chain < of (C2(M),t). Since such a
propagating chain represents the linking number, ©(M, t) can be thought of as the cube of the linking number with respect to t.
When t varies continuously, ©(M, t) varies continuously in Q so that ©(M, t) is an invariant of the homotopy class of t.
2.5. Parallelisations of 3-manifolds and Pontrjagin classes
In this subsection, M denotes a smooth, compact oriented 3-manifold with possible boundary dM. Recall that such a 3-manifold is parallelizable.
Let GIL (R3) denote the group of orientation-preserving linear isomorphisms of R3. Let C0((M,dM),(GL+ (R3),1)) denote the set of maps g: (M,dM) ^ (GL+ (R3),1) from M to GL+ (R3) that send dM to the unit 1 of GL+ (R3). Let [(M, dM),(GL+ (R3),1)] denote the group of homoto-py classes of such maps, with the group structure induced by the multiplication of maps, using the multiplication in GL+ (R3). For a map g in C0((M,dM),(GL+ (R3),1)), set
(g): M x R3 ^ M x R3,
(x, y) ^ (x, g(x)(y)). Let tm : M x R3 ^ TM be a parallelization of M. Then any parallelization t of M that coincides with tm on dM reads t = tm ° yR(g) for some geC0((M,dM),(GL+ (R3),1)).
Thus, fixing tm identifies the set of homotopy classes of parallelizations of M fixed on dM with the group [(M, dM),(GL+ (R3),1)]. Since GL+ (R3) deformation retracts onto the group SO(3) of orientation-preserving linear isometries of R3, [(M, dM ),(GL+ (R3),1)] is isomorphic to [(M, dM ),(SO(3),1)].
See S3 as B3/dB3 where B3 is the standard ball of radius 2n of R3 seen as ([0,2n]xS2)/(0~{0}xS2). Let p: B3 ^ SO(3) map (6e[0,2n],v eS2) to the rotation p(9, v) with axis directed by v and with angle 0. This map induces the double covering p: S3 ^ SO(3), which orients SO(3) and
which allows one to deduce the first three homotopy groups of SO(3) from the ones of S3. They are nI(SO(3)) = Z / 2Z, tc2(SO(3)) = 0 and n3(SO(3)) = Z[p]. For v e S2, n1(SO(3)) is generated by the class of the loop that maps exp(z'0) e S1 to the rotation p(9, v).
Note that a map g from (M, dM) to (<SÜ(3),1) has a degree deg(g), which may be defined as the differential degree at a regular value (different from 1) of g. It can also be defined homo-logically, by H3(g)[M,dM] = deg(g)[SO(3),1].
The following theorem is proved in Section 6.
Theorem 2. For any smooth compact connected oriented 3-manifold M, the group [(M, dM),(SO(3),1)] is abelian, and the degree deg: [(M, dM),(50(3),1)] ^Z is a group homo-morphism, which induces an isomorphism deg: [(M, dM),(<S0(3),1)] ®Z Q ^ Q. When dM = 0, (resp. when dM = S2), there exists a canonical map p1 from the set of homotopy classes of parallelizations of M (resp. that coincide with — near S2) such that for any map g in C0 ((M, dM),(SO(3),1)), for any trivialization t of TM
p1(t ° yR(g)) - p1(T)=2deg(g).
The definition of the map p1 is given in Subsection 6.5, it involves relative Pontrjagin classes. When dM = 0, the map p1 coincides with the map h that is studied by Kirby and Melvin in [7] under the name Hirzebruch defect. See also [8. § 3.1].
Since [(M, dM),(5"0(3),1)] is abelian, the set of parallelizations of M that are fixed on dM is an affine space with translation group [(M, dM ),(5"0(3),1)].
Recall that p: B3 ^ SO(3) maps (0e [0,2n], v e S2) to the rotation with axis directed by v and with angle 0. Let M be an oriented connected 3-manifold with possible boundary. For a ball B3 embedded in M, let pM (B3) e C0 ((M, dM),(SO(3),1)) be a (continuous) map that coincides with p on B3 and that maps the complement of B3 to the unit of SO(3). The homotopy class of pM(B3) is well-defined.
Lemma 10. deg (pM (B3)) = 2.
Proof. Exercise. □
2.6. Defining a Q-sphere invariant from ®
Recall that an asymptotic rational homology M3 is a pair (M, t) where M is 3-manifold that reads as the union over ]1,2] x S2 of a rational homology ball BM and the complement Blx of the unit ball of R3, and that is equipped with an asymptotically standard parallelization t.
In this subsection, we prove the following proposition.
Proposition 3. Let (M,t) be an asymptotic rational homology R3. For any map g in C0 ((BM, BM Blm ),(^0(3),1)) trivially extended to M,
©(M, t o (g)) - 0(M, t) = \ deg(g).
Theorem 2 allows us to derive the following corollary from Proposition 3.
Corollary 1. ©(M) = ©(M,t)- ^p(t) is an invariant of Q-spheres.
Lemma 11. ©(M, t o yR (g)) -©(M,t) is independent of t. Set
©'(g) = &(M, t o ¥R (g)) - &(M, t).
Then is a homomorphism from [(BM,BM ^ ),(^0(3),1)] to Q.
Proof. For d = a, b or c, the propagator Vd of (C2(M),t) can be assumed to be a product [-1,0] x p-^ (d) on a collar [-1,0] x UBM of UBM in C2(M). Since #3([-1,0] x UBM;Q) = 0,
(d([-1,0] x p-1BM (d)) \ (0 x p-1BM (d))) u (0 x p-;r(?)
\UBM (d))
The chains Ga, Gb and Gc can be assumed to be transverse. Construct the propagator Vd(g) of (C2(M),- o vr(g)) from Vd by replacing [-1,0]x p-^ (d) by Gá on [-1,0] x UBM . Then
0(M, t ° yR(g))- ®(M,x) is equal to (Ga,Gb,Gc)[_i
allows us to see that 0(M,t ° yR(g)) -0(M,t) is independent of t. Then it is easy to observe that 0' is a homomorphism from [(BM, dBM ),(SO(3),1)] to Q. □
According to Theorem 2 and to Lemma 10, it suffices to prove that 0' (pM (B3)) = 1 in order to prove Proposition 3. It is easy to see that 0'(pM (B3)) = 0'(p) .Thus,we are left with the proof of the following lemma. Lemma 12. 0' (p) = 1.
Again, see B3 as ([0,2n] x S2) / (0 - {0} x S2). We first prove the following lemma: Lemma 13. Let a be the North Pole. The point (-a) is regular for the map
p a : B3 ^ S2
m ^ p(m)(a)
and its preimage (cooriented by S2 via pa) is the knot La = {n} x E, where E is the equator that bounds the Southern Hemisphere. Proof. It is easy to see that p-1 (-a) = ±{n} x E .
Let x e {7t} x E . When m moves along the great circle that contains a and x from x towards (-a) in {n} x S2, p(m)(a) moves from (-a) in the same direction, which will be the direction of the tangent vector v1 of S2 at (-a), counterclockwise in our picture, where x is on the left. Then in our picture, S2 is oriented at (-a) by v1 and by the tangent vector v2 at (-a) towards us. In order to move p(6,v)(a) in the v2 direction, one increases 0 so that La is cooriented and oriented like in the figure. □ Proof of Lemma 12. We use the notation of the proof of Lemma 11 and we construct an
t
explicit Ga in [-1,0] x UB3 =[-1,0] x B3 x S2.
When p(m)(a) a, there is a unique geodesic arc [a, p(m)(a)\ with length (£ e[0, n[) from a to p(m)(a) = pa(m). For t e [0,1], let Xt(m) e [a,pa(m)] be such that the length of [X0(m) = a,Xt(m)] is t£ . This defines Xt on (M \La), Xl(m) = pa(m). Let us show how the definition of Xt smoothly extends on the manifold B£(B3, La) obtained from B3 by blowing up La.
The map pa maps the normal bundle to La to a disk of S2 around (-a), by an orientation-preserving diffeomorphism on every fiber (near the origin). In particular, pa induces a map pa from the unit normal bundle to La to the unit normal bundle to (-a) in S2, which preserves the orientation of the fibers. Then for an element y of the unit normal bundle to La in M, define Xt(y) as before on the half great circle [a, -a]p ( ) from a to (-a) that is tangent to pa(-y) at (-a) (so that pa(-y) is an outward normal to [a,-a]p (-y) at (-a)). This extends the definition of Xt, continuously.
The whole sphere is covered with degree (-1) by the image of ([0,1] x UVx(La)), where the fiber UVx (La) of the unit normal bundle to La is oriented as the boundary of a disk in
the fiber of the normal bundle. Let Gh(a) be the closure of I u 3 (m,X (m)) in UB3
h ^ ie[0,l],me( B \ La ' >)
= Ute[mmeBl(B3^a) (PB3 mXt(m)\ Then
dGh = -(B3 xa) + UmeB3 (m,pa(m)) + u(e[ai](-dBt(S3,La))
where (-dB£(S3,La)) is oriented like dN(La) so that the last summand reads (-La xS2) because the sphere is covered with degree (-1) by the image of ([0,1] x U Vx (La)). Let Da be a disk bounded by La in B3. Set G(a) = Gh(a) + Da x S2 so that
dG(a) = -(B3 xa) + u 3(m, pa (m)).
meB
Now let i be the endomorphism of UB3 over B3 that maps a unit vector to the opposite one. Set Ga =[-1,-2/3] xB3 xa + {-2/3} xG(a) + [-2/3,0] xu^m,pa(m)) and G_a = [-1,-1 /3] x
2?3x(-a) + {-1 / 3} x i(G(a)) + [-1/3,0] x u^3 (m,p(m)(-a)). Then Ga n G-a = [-2 / 3,-1/3] xLa x (-a) + {-2/3} xDa x (-a) - {-1/3} x umeD (m,Pa(m)). Finally, ®'(p) is the algebraic intersection of Ga n G_a with Pc (p) in C2(M). This intersection coincides with the algebraic intersection of Ga nG-a with any propagator of (C2(M),t) according to Lemma 8. Therefore ®'(p) =
(Pa' Ga n G-a )[_1fi]xS2 B = _ deg a(Pa ■ Da ^ SX The orientation of La allows us to choose (-Da) as the Northern Hemisphere, the image of this hemisphere under pa covers the sphere with degree 1 so that ®'(p) = 1. □
3. An introduction to finite type invariants
This section contains the needed background from the theory of finite type invariants. It allows us to introduce the target space generated by Feynman-Jacobi diagrams, for the general invariants presented in Section 4, in a progressive way.
Theories of finite type invariants are useful to characterize invariants. Such a theory allowed Greg Kuperberg and Dylan Thurston to identify 0 /6 with the Casson invariant X for integer homology 3-spheres, in [9]. The invariant X was defined by Casson in 1984 as an algebraic count of conjugacy classes of irreducible representations from n1(M) to SU(2). See [10-12]. The Kuperberg — Thurston result above was generalized to the case of rational homology 3-spheres in [13; Theorem 2.6 and Corollary 6.14]. Thus, for any rational homology 3-sphere M, 0(M) = 6X(M), where X is the Walker generalization of the Casson invariant to rational homology 3-spheres, which is normalized like in [10-12] for integer homology 3-spheres, and like XW
for rational homology 3-spheres with respect to the Walker normalisation XW of [14].
For invariants of knots and links in M3, the base of the theory of finite type invariants was mainly established by Bar-Natan in [15]. A more complete review of this theory has been written by Chmutov, Duzhin and Mostovoy in [16]. For integer homology 3-spheres, the theory was started by Ohtsuki in [17] and further developed by Goussarov, Habiro, Le and others. See [18-20]. Delphine Moussard developed a theory of finite type invariants for rational homology 3-spheres in [21]. Her suitable theory is based on the Lagrangian-preserving surgeries defined below.
3.1. Lagrangian-preserving surgeries
Definition 2. An integer (resp. rational) homology handlebody of genus g is a compact oriented 3-manifold A that has the same integral (resp. rational) homology as the usual solid handlebody Hg of Fig. 1.
Fig. 1. The handlebody hg
Exercise 1. Show that if A is a rational homology handlebody of genus g, then dA is a genus g surface.
The Lagrangian CA of a compact 3-manifold A is the kernel of the map induced by the inclusion from Hj(dA;Q) to H1(dA;Q).
In Fig. 1, the Lagrangian of Hg is freely generated by the classes of the curves a. Definition 3. An integral (resp. rational) Lagrangian-Preserving (or LP) surgery (A '/A) is the replacement of an integral (resp. rational) homology handlebody A embedded in the interior of a 3-manifold M by another such A' whose boundary is identified with dA by an orientation-preserving diffeomorphism that sends CA to CA,. The manifold M(A'/A) obtained by such an LP-surgery reads M{A '/A) = (M\ Int(A)) <ugA A '.(Thisonlydefinesthe topological structure of M(A'/A), but we equip M(A'/A) with its unique smooth structure.)
Lemma 14. If (A'/A) is an integral (resp. rational) LP-surgery, then the homology of M(A' /A) with Z-coefficients (resp. with Q-coefficients) is canonically isomorphic to H* (M;Z) (resp. to H*(M;Q)). If M is a Q-sphere,if (A'/A) is a rational LP-surgery,and if (J,K) is a two-component link of M \ A,then the linking number of J and K in M and the linking number of J and K in M( A /A) coincide. Proof. Exercise. □
3.2. Definition of finite type invariants
Let K = Q or M. A K-valued invariant of oriented 3-manifolds is a function from the set
n
of 3-manifolds, considered up to orientation-preserving diffeomorphisms to K. Let J JA',' de-
i=i
note a disjoint union of n circles, where each S) is a copy of S1. Here, an n-component link in a 3-manifold M is an equivalence class of smooth embeddings L ^ ^ M under the
equivalence relation that identifies two embeddings L and L' if and only if there is an orientation-preserving diffeomorphism h of M such that h(L) = L'. A knot is a one-component link. A link invariant (resp. a knot invariant) is a function of links (resp. knots). For example, ® is an invariant of Q-spheres and the linking number is a rational invariant of two-component links in rational homology 3-spheres.
In order to study a function, it is usual to study its derivative, and the derivative of its derivative The derivative of a function is defined from its variations. For a function f from
Zd = ©f=1 Zei to K, one can define its first order derivatives f : Zd ^ K by
de,
f (z) = f (z + ) - f (z)
de,
and check that all the first order derivatives of f vanish if and only if f is constant. Inductively define an n-order derivative as a first order derivative of an (n-1)-order derivative for a positive integer n. Then it can be checked that all the (n + 1)-order derivatives of a function vanish if and only if f is a polynomial of degree not greater than n. In order to study topolog-ical invariants, we can similarly study their variations under simple operations.
Below, X denotes one of the following sets:
• Zd;
• the set K of knots in M3, the set Kn of n-component links in R3;
• the set M of Z-spheres, the set MQ of Q-spheres.
And O(X) denotes a set of simple operations acting on some elements of X.
For X = Zd, O(X) will be made of the operations (z ^ z ± e{).
For knots or links in R3, the simple operations will be crossing changes. A crossing change ball of a link L is a ball B of the ambient space, where L n B is a disjoint union of two arcs aj and a2 properly embedded in B, and there exist two disjoint topological disks D1 and D2 embedded in B, such that, for i e {1,2}, a; and (3Di \ a,.) After an isotopy, the
projection of (B, apa2) reads ;'X) or (X} (the corresponding pairs (ball,arcs) are isomorphic, but they are regarded in different ways), a crossing change is a change that does not change L outside B and that modifies L inside B by a local move (i'X^'-'X)) or ('X^'^X)). For the move (i'X^(X}), the crossing change is positive, it is O'XW'-'X') negative for the move ('fX^'-X^.
For integer (resp. rational) homology 3-spheres, the simple operations will be integral (resp. rational) LP-surgeries of genus 3.
Say that crossing changes are disjoint if they sit inside disjoint 3-balls. Say that LP-sur-geries (A'/A) and (B'/B) in a manifold M are disjoint if A and B are disjoint in M. Two operations on Zd are always disjoint (even if they look identical). In particular, disjoint operations commute, (their result does not depend on which one is performed first). Let n = {1,2,...,n}. Consider the vector space X) freely generated by X over K. For an element x of X and n pairwise disjoint operations oJ,...,on acting on x, define
[x;0l,..„] = £(-1)#Ix((oi),e/) e f0(X) I c n
where x((oi)ieI) denotes the element of X obtained by performing the operations oi for i e I on x. Then define Fn(X) as the K-subspace of F0(X) generated by the [x;on], for all x e X equipped with n pairwise disjoint simple operations. Since
[ x; Oj,..., on, o„+1] = [ x; o^,..., on ]- [ x(oB+1); on ], Fn+l(X) c Tn(X), for all n e N.
Definition 4. A K-valued function f on X, uniquely extends as a K-linear map of
F0(X )* = Hom(T0( X); K),
which is still denoted by f. For an integer n e N, the invariant (or function) f is of degree < n if and only if f(fr„+1 (X)) = 0. The degree of such an invariant is the smallest integer n e N such that f(Tn+I (X)) = 0. An invariant is of finite type if it is of degree n for some n e N. This definition depends on the chosen set of operations O(X). We fixed our choices for our sets X, but other choices could lead to different notions. See [18].
Let Xn (X ) = (T0(X)/ Fn+x(X ))* be the space of invariants of degree at most n. Of course, for all n e N, ln (X) c ln+j (X).
Example 2. Xn (Zd) is the space of polynomials of degree at most n on Zd. (Exercise.)
Lemma 15. Any n-component link in R3 can be transformed to the trivial n-component link below by a finite number of disjoint crossing changes.
Proof. Let L be an n-component link in R3. Since R3 is simply connected, there is a homot-
n
opy that carries L to the trivial link. Such a homotopy h: [0,1] x J ^ M3 can be chosen, so
i=i
that h(t, •) is an embedding except for finitely many times t, 0 < t1<... < tt < tM < tk <1 where hit^, •) is an immersion with one double point and no other multiple points, and the link h(t, •) changes exactly by a crossing change when t crosses a t. (For an alternative elementary proof of this lemma, see [22. Subsection 7.1].) □
In particular, a degree 0 invariant of n-component links of R3 must be constant, since it is not allowed to vary under a crossing change.
Exercise 2. 1. Check that X, (K) = Kc0, where c0 is the constant map that maps any knot to 1.
2. Check that the linking number is a degree 1 invariant of 2-component links of R3.
3. Check that lI(K2) = Kc0 © Klk, where c0 is the constant map that maps any two-component link to 1.
3.3. Introduction to chord diagrams
Let f be a knot invariant of degree at most n. We want to evaluate f([K;oJ,...,on]) where the oi are disjoint negative crossing changes
^ ^ ^
to be performed on a knot K. Such a [K,ol,...,on] is usually represented as a singular knot with n double points that is an immersion of a circle with n transverse double points where
each double point can be desingularized in two ways, the positive one and the negative one ^ and K is obtained from the singular knot by desingularizing all the crossings in the
positive way, which is
in our example. Note that the sign of the desingularization is
defined from the orientation of the ambient space.
Define the chord diagram c^,...,on]) associated to [K;c^,...,on] as follows. Draw the preimage of the associated singular knot with n double points as an oriented dashed circle equipped with the 2n preimages of the double points and join the pairs of preimages of a double point by a plain segment called a chord.
/ \ Formally, a chord diagram with n chords is a cyclic order of the 2n
r I \f ) = <f9 ends of the n chords, up to a permutation of the chords and up to exchanging the two ends of a chord.
Lemma 16. When f is a knot invariant of degree at most n, f([K;c^,...,on]) only depends on r([K;ol5...,on]).
Proof. Since f is of degree n, f([K;ol,...,on]) is invariant under a crossing change outside the balls of the o, that is outside the double points of the associated singular knot. Therefore, f ([K;oj,...,on]) only depends on the cyclic order of the 2n arcs involved in the oi on K. □
Let Vn be the K-vector space freely generated by the n chord diagrams on 51. Then
Lemma 17. The map §n from Vn to
that maps r to some [K ; oj,..., on ] whose diagram
is r is well-defined and surjective. Proof. Use the arguments of the proof of Lemma 16.
For example, (j)3 j ) =
The kernel of the composition of and the restriction below
^p(E)
V y
y ,
(V)
V rnvi(K) y
*
V
is ln I(K). Thus, —"^ K injects into T>*n and Tn(K) is finite dimensional for all n. Furthermore,
^n-l(V)
= Hom
fSEL.
v
An isolated chord in a chord diagram is a chord between two points of S1 that are consecutive on the circle.
Lemma 18. Let D be a diagram on S1 that contains an isolated chord. Then §n (D) = 0. Let D1, D2, D3, D4 be four n-chord diagrams that are identical outside three portions of circles where they look like:
□
then (-D1 + D' + T>3 - D4) = 0.
Proof. For the first assertion, observe that <|)w( ) = — F°r the second one, see
[22. Lemma 2.21], for example. □
Let Vn denote the quotient of Vn by the four-term relation, which is the quotient of Vn by the vector space generated by the (-D1 + D2 + D3 - D4) for all the 4-tuples (D1,D2,D3,D4) as above. Call (1T) the relation that identifies a diagram with an isolated chord with 0 so that Vn /(IT) is the quotient of Vn by the vector space generated by diagrams with an isolated chord.
According to Lemma 18 above, the map induces a map
Tn (E)
1: Vn/(IT) ^
•^i(K)
The fundamental theorem of Vassiliev invariants (which are finite type knot invariants) can now be stated.
Theorem 3. There exists a family of linear maps (Zf : ^ Vn) such that:
K (?„+1(K)) = 0;
Z'K induces the inverse of from
to Vn /(IT);
In particular ^ = Vn / QT) and J"= (Vn / (1T))*.
^n-l(E)
This theorem has been proved by Kontsevich and Bar-Natan in [15] using the Kontsevich integral ZK = (Zn)BeN described in [23] and in [16. Chapter 8], for K = M. It is also true when K = Q.
Remark 1. The Kontsevich integral has been generalized to a functor from the category of framed tangles to a category of Jacobi diagrams by Le and Murakami in [24]. Le and Murakami showed how to derive the i. e.Turaev quantum invariants of framed links in M3 defined in [25; 26] from their functor, in [24. Theorem 10].
3.4. More spaces of diagrams
Definition 5. A uni-trivalent graph r is a 6-tuple (H(r), E(r),^(r),T(r), pE, pV) where _H(r), £(r), U(T) and T(r) are finite sets, which are called the set of half-edges of r, the set of edges of r, the set of univalent vertices of r and the set of trivalent vertices of r, respectively, pE : H(r) ^ E(r) is a two-to-one map (every element of £(r) has two preimages under pE) and pV : H(r) ^ U(r^ JT(r) is a map such that every element of U(r) has one preimage under pV and every element of T(r) has three preimages under pV, up to isomorphism. In other words, r is a set H(r) equipped with two partitions, a partition into pairs (induced by pE), and a partition into singletons and triples (induced by pV), up to the bijections that preserve the partitions. These bijections are the automorphisms of r.
Definition 6. Let C be an oriented one-manifold. A Jacobi diagram r with support C, also called Jacobi diagram on C, is a finite uni-trivalent graph r equipped with an isotopy class of injections ip of the set U(r) of univalent vertices of r into the interior of C. A vertex-orientation of a Jacobi diagram r is an orientation of every trivalent vertex of r, which is a cyclic order on the set of the three half-edges which meet at this vertex. A Jacobi diagram is oriented if it is equipped with a vertex-orientation.
Such an oriented Jacobi diagram r is represented by a planar immersion of r u C where the univalent vertices of U(r) are located at their images under ir, the one-manifold C is repre-
sented by dashed lines, whereas the diagram r is plain. The vertices are represented by big points. The local orientation of a vertex is represented by the counterclockwise order of the three half-edges that meet at it.
Here is an example of a picture of a Jacobi diagram r on the disjoint union M = S|,S1 of
The degree of such a diagram is half the number of all the vertices of r.
Of course, a chord diagram of Vn is a degree n Jacobi diagram on S1 without trivalent vertices.
Let T>1 (C) denote the K-vector space generated by the degree n oriented Jacobi diagrams on C.
v[(sl) = k(j> © k(j> © k:0- © © k($>.
Let Vtn(C) denote the quotient of Vtn(C) by the following relations AS, Jacobi and STU:
As before, each of these relations relate oriented Jacobi diagrams which are identical outside the pictures where they are like in the pictures.
Remark 2. Lie algebras provide nontrivial linear maps, called weight systems from Vn(C) to K, see [15] and [22. Section 6]. In the weight system constructions, the Jacobi relation for the Lie bracket ensures that the maps defined for oriented Jacobi diagrams factor through the Jacobi relation. In [27], Pierre Vogel proved that the maps associated to Lie (super)algebras are sufficient to detect nontrivial elements of Vtn (C) until degree 15, and he exhibited a non trivial element of T>16(0) that cannot be detected by such maps. The Jacobi relation was originally called IHX by Bar-Natan in [15] because, up to AS, it can be written as ^ _ {—f _ X •
Set V„(0) = V„(0-K) = V:(0).
When C , let Vn(C) = Vn(C;K) denote the quotient of Vn(C) = Vn(C;K) by the vector space generated by the diagrams that have at least one connected component without univalent vertices. Then Vn (C) is generated by the oriented Jacobi diagrams whose (plain) connected components contain at least one univalent vertex.
Proposition 4. The natural map from Vn to Vn(Sinduces an isomorphism from Vn to Vn (Sl).
Sketch of proof. The natural map from Vn to Vn(Sfactors though 4T since, according to
STU,
in Vtn (SSince STU allows us to inductively write any oriented Jacobi diagram whose connected components contain at least a univalent vertex as a combination of chord diagrams, the induced map from Vn to Vn (Sis surjective. In order to prove injectivity, one constructs an inverse map. See [22. Subsection 3.4]. □
The Le fundamental theorem on finite type invariants of Z-spheres is the following one.
Theorem 4. There exists a family (ZL° : M) ^ Vn(0)) of linear maps such that:
• (f2n+1(M)) = 0; "eN
(M)
induces an isomorphism from
to Vn (0);
n
•m(M)
•n (M)
= {0}.
In particular •(M) = Vn (0) and J2"M = V* (0).
This theorem has been proved by Le [20] using the Le-Murakami-Ohtsuki invariant zimo = (zlmo) ^ of [28]. As explained in [29], this LMO invariant contains the quantum Wit-ten-Reshetikhin-invariants of rational homology 3-spheres defined in [30].
In [21], Delphine Moussard obtained a similar fundamental theorem for finite type invariants of Q-spheres using the configuration space integral ZKKT described in [9; 31] and in Theorem 5 below.
As in the knot case, the hardest part of these theorems is the construction of an invariant Z = (Zn )neN that has the required properties. We will define such an invariant by "counting Jacobi diagram configurations" in Subsection 4.3 and explain why it satisfies the required so-called universality properties in Subsection 4.4.
3.5. Multiplying diagrams
Se-t V'(C) = n(C) and V(C) = n„eKV (C).
Assume that a one-manifold C is decomposed as a union of two one-manifolds C = Cx u C2 whose interiors in C do not intersect. Define the product associated to this decomposition:
V (Cj) x V (C2) ^ V (C)
as the continuous bilinear map which maps ([rj,[r2]) to |r, |]T2], if ri is a diagram with support C1 and if r2 is a diagram with support C2, where I | Jr2 denotes their disjoint union.
In particular, the disjoint union of diagrams turns V(0) into a commutative algebra graded by the degree, and it turns Vt(C) into a V(0)-module, for any 1-dimensional manifold C.
An orientation-preserving diffeomorphism from a manifold C to another one C induces an isomorphism from Vn (C) to Vn(C'), for all n.
Let I = [0,1] be the compact oriented interval. If I = C, and if we identify I with Cx = [0,1/2] and with C2 = [1/ 2,1] with respect to the orientation, then the above process turns V(I) into an algebra where the elements with non-zero degree zero part admit an inverse.
Proposition 5. The algebra V([0,1]) is commutative. The projection from [0,1] to S1 = [0,1] / (0 ^ 1) induces an isomorphism from Vn([0,1]) to Vn(S1) for all n, so that V(S inherits a commutative algebra structure from this isomorphism. The choice of a connected component Cj of C equips V(C) with an V([0,1])-module structure #,induced by the inclusion from [0,1] to a little part of Cj outside the vertices,and the insertion of diagrams with support [0,1] there.
In order to prove this proposition, we present a useful trick in diagram spaces.
First adopt a convention. So far, in a diagram picture, or in a chord diagram picture, the plain edge of a univalent vertex, has always been attached on the left-hand side of the oriented one-manifold. Now, if k plain edges are attached on the other side on a diagram picture, then we agree that the corresponding represented element of V (M) is (-1)* times the underlying diagram. With this convention, we have the new antisymmetry relation in vtn (M):
-_!-> + -(-}-> = 0> and we can draw the STU relation like the Jacobi relation:
Lemma 19. Let r be a Jacobi diagram with support C. Assume that r u C is immersed in the plane so that r u C meets an open annulus A embedded in the plane exactly along n + 1 embedded arcs a4,a2,... ,an and $,and one vertex v so that:
1) the a{ may be dashed or plain,they run from a boundary component of A to the other one;
2) p is a plain arc which runs from the boundary of A to v ea,;
3) the bounded component D of the complement of A does not contain a boundary point of C. Let r be the diagram obtained from r by attaching the endpoint v of p to a{ instead of ax
on the same side, where the side of an arc is its side when going from the outside boundary
n
component of A to the inside one dD. Then ^r. = 0 in V (C).
¿=i
Examples 3.
Proof. The second example shows that the STU relation is equivalent to this relation when the bounded component D of M2 \A intersects r in the neighborhood of a univalent vertex on C. Similarly, the Jacobi relation is easily seen as given by this relation when D intersects r in the neighborhood of a trivalent vertex. Also note that AS corresponds to the case when D intersects r along a dashed or plain arc. Now for the Bar-Natan [15. Lemma 3.1] proof. See also [27. Lemma 3.3]. Assume without loss that v is always attached on the right-hand-side of the a's. Add to the sum the trivial (by Jacobi and STU) contribution of the sum of the diagrams obtained from ri by attaching v to each of the three (dashed or plain) half-edges of each vertex w of r u C in D on the left-hand side when the half-edges are oriented towards w. Now, group the terms of the obtained sum by edges of r u C where v is attached, and observe that the sum is zero edge by edge by AS.
Proof of Proposition 5. To each choice of a connected component C ■ of C, we associate an v(l)-module structure on V(C), which is given by the continuous bilinear map:
V(I) x V(C) ^V(C)
such that: if r' is a diagram with support C and if r is a diagram with support I, then ([r],[r ']) is mapped to the class of the diagram obtained by inserting r along Cj. outside the vertices of r, according to the given orientation. For example,
As shown in the first example that illustrates Lemma 19, the independence of the choice of the insertion locus is a consequence of Lemma 19, where ri is the disjoint union r]Jr ' and r1 intersects D along FuI. This also proves that V(I) is a commutative algebra. Since the morphism from V(I) to V(S1) induced by the identification of the two endpoints of I amounts to quotient out V(I) by the relation that identifies two diagrams that are obtained from one another by moving the nearest univalent vertex to an endpoint of I near the other endpoint, a similar application of Lemma 19 also proves that this morphism is an isomorphism from V(I) to v(sl). (In this application, p comes from the inside boundary of the annulus.) □
4. Configuration space construction of universal finite type invariants
In this section, we finally describe the promised invariants, which generalize both the linking number and These invariants count configurations of Jacobi diagrams with support some link, in an asymptotic rational homology M3. In Subsection 4.1, we introduce the relevant configuration spaces. In Subsection 4.2, we define integrals over these spaces from propagating forms. The wanted invariants are obtained by combining these integrals in Subsection 4.3. These integrals will be expressed in terms of algebraic intersections, which involve propagating chains, in Subsection 5.3. Important universality properties of the constructed invariants are presented in Subsection 4.4.
4.1. Configuration spaces of links in 3-manifolds
Let (M, t) be an asymptotic rational homology M3. Let C be a disjoint union of k circles 5?, i e k , and let L : C ^ M denote a C" embedding from C to M. Let r be a Jacobi diagram with support C. Let U = U(r) denote the set of univalent vertices of r, and let T = T(r) denote the set of trivalent vertices of r. A configuration of r is an embedding c: Uu T ^ M whose restriction c^U to U may be written as L ° j for some injection j : U ^ C in the given isotopy class [ir ] of embeddings of U into the interior of
C. Let C(L; r) = {c : U u T ^ M; 3j e i ], c^ = L ° j} denote the set of these configurations.
In C(L; r), the univalent vertices move along L(C) while the trivalent vertices move in the
ambient space, and C(L; r) is naturally an open submanifold of CU x MT.
An orientation of a set of cardinality at least 2 is a total order of its elements up to an even permutation.
Cut each edge of r into two half-edges. When an edge is oriented, define its first half-edge and its second one, so that following the orientation of the edge, the first half-edge is met first. Recall that H(r) denotes the set of half-edges of r.
Lemma 20. When r is equipped with a vertex-orientation, orientations of the manifold C(L; r) are in canonical one-to-one correspondence with orientations of the set H(r).
Proof. Since C(L; r) is naturally an open submanifold of CU x MT, it inherits M#U+3#T-valued charts from M-valued orientation-preserving charts of C and M3-valued orientation-preserving charts of M. In order to define the orientation of M#U+3#T, one must identify its factors and order them (up to even permutation). Each of the factors may be labeled by an element of H(r): the M-valued local coordinate of an element of C corresponding to the image under j of an element of U sits in the factor labeled by the half-edge of U; the 3 cyclically ordered (by the orientation of M) M-valued local coordinates of the image under a configuration c of an element of T live in the factors labeled by the three half-edges that are cyclically ordered by the vertex-orientation of r, so that the cyclic orders match. □
The dimension of C(L; r) is #U(r) + 3# T(r) = 2# E(r) where E = E(r) denotes the set of
1
edges of r. Since n = n(r) = -(#U(r) + # T(r)), #E(r) = 3n - #U(r).
4.2. Configuration space integrals
A numbered degree n Jacobi diagram is a degree n Jacobi diagram r whose edges are oriented, equipped with an injection jE : E(r)^ 3n. Such an injection numbers the edges. Note that this injection is a bijection when U(r) is empty. Let V^(C) denote the set of numbered degree n Jacobi diagrams with support C without looped edges like -O.
Let r be a numbered degree n Jacobi diagram. The orientations of the edges of r induce the following orientation of the set H(r) of half-edges of r: order E(r) arbitrarily, and order the
half-edges as (first half-edge of the first edge, second half-edge of the first edge,..., second half-edge of the last edge). The induced orientation is called the edge-orientation of H(r). Note that it does not depend on the order of E(r). Thus, as soon as r is equipped with a vertex-orientation o(r), the edge-orientation of r orients C(L; r).
An edge e oriented from a vertex vl to a vertex v2 of r induces the following canonical map
pe : C(L; r) ^ C2(M),
c ^ (c(v1), c(v2)).
For any i e 3n, let ©(i) be a propagating form of (C2(M), t). Define I(r, o(r),(©(i))ie3n) as
I(r,o(r),(©(i))) = f ^ A p*(©(j(e)))
where C(L; r) is equipped with the orientation induced by the vertex-orientation o(r) and by the edge-orientation of r.
The convergence of this integral is a consequence of the following proposition, which will be proved in Subsection 5.1.
Proposition 6. There exists a smooth compactification C(L; r) of C(L;r) where the maps pe smoothly extend.
According to this proposition, aeeE(rpC(©(jE(e))) smoothly extends to C(L;r), and
i(C(L;r),o(r))A ee^^E (e))) is equal to j(C(L;r),o(r)) A eeE^*(©( jE (e))).
Examples 1. For any three propagating forms ©(1), ©(2) and ©(3) of (C2(M), t), /(5/t;w>5/, (©(¿)),:e3) = ik(K;,KX /(0,(®(i»iea) = ®(M,T)
for any numbering of the (plain) diagrams (exercise).
Let us now study the case of /(t^?5-1,(ro(i));e3), which depends on the chosen propagating forms, and on the diagram numbering.
A dilation is a homothety with positive ratio.
Let U+Kj denote the fiber space over Kj made of the tangent vectors to the knot Kj of M that orient Kj, up to dilation. The fiber of U+Kj is made of one point, so that the total space of this unit positive tangent bundle to Kj is Kj. Let U-Kj denote the fiber space over Kj made of the opposite tangent vectors to Kj, up to dilation.
For a knot Kj in M , define the two-point configuration space C(Kf, ) as
{(/C(?),/C(?cxp(ie)));(?,e) e S'xIO^ttQ. Let C. = C(K.; ¿r}S-) be the closure of in CXM). This closure is diffeomor-
J J ~ J J ~ J ^
phic to 51' x [0?27r] where S1 x 0 is identified with U+K , 5,1x{27i} is identified with U K and dC(Kj-^}Sj) = U+Kj-UKj.
Lemma 21. For any ie 3, let ©(¿) and (o'(i) = (o(i) + dr\(i) be propagating forms of (C2(M),t), where n(i) is a one-form on C2(M). Then
K^rs;, (©'(¿)),:ea) - (©(¿)),:ea) = LK,^k)-LK,^k)-
Proof. Apply the Stokes theorem to Jc (o>'(k) - o>(k)) =Jc dr\(k). □
Exercise 2. Find a knot Kj of M3 and a form r|(k) of C2(M3) such that the right-hand side of Lemma 21 does not vanish. (Use Lemma 9, hints can be found in Subsection 5.2.)
Say that a propagating form © of (C2(M),t) is homogeneous if its restriction to dC2(M) is p*(© 2) for the homogeneous volume form © 2 of S2 of total volume 1.
Lemma 22. For any i e 3, let ©(i) be a homogeneous propagating form of (C2(M),t). Then
©(¿))(e3) does not depend on the choices of the (o(i),it is denoted by Ie(Kj,x) . Proof. Apply Lemma 21 with nA = 0 , so that n(k) = 0 in Lemma21. □
4.3. An invariant for links in Q-spheres from configuration spaces
Let K = R. Let [r, o(r)] denote the class in Vtn (C) of a numbered Jacobi diagram r of V (C) equipped with a vertex-orientation o(r), then I(r, o(r),(©(i))ie3n)[r, o(r)] e V(C) is independent of the orientation of o(r), it will be simply denoted by I(r,(©(i))ie3n )[r].
Theorem 5. Let (M, t) be an asymptotic rational homology R3. Let L : JJk,=iSlj ^M be an embedding. For any i e 3n,let ©(i) be a homogeneous propagating form of (C2(M),t) . The sum
Fe»n (C) (3n)!2
in V (^ k._Slj) is independent of the chosen ©(i). It only depends on the diffeomorphism class of (M, L),on p1(t) and on the i0(Kj, t) ,for the components Kj of L. It is denoted by Zn(L,M,t). More precisely, set Z(L,M,t) = (Zn(L,M,T))neN e Vt (^k._Sl). There exist two constants
1
a e V(S4; Q) and p e V(0; Q) such that the product of exp(- — p1(t)P) by
( exp(-I0 (Kj, T)a) # y) Z(L, M, t) ,
j=i
where exp(-I0 (Kj)a) acts on Z(L, M, z),on the copy sj of Sl as indicated by the subscript j,only depends on the diffeomorphism class of (M,L). It is denoted by Z(L,M),
Z(L, M) e Vt (JJSj; Q).
j=i
Furthermore,if M = R3,then the projection Zu(L,S3) of Z(L,S3) on V(| J ^Sy) is a universal finite type invariant of links in R3, i. e. ZU satisfies the properties stated for ZK in Theorem 3. It is the configuration space invariant studied by Altschbler,Freidel [32],Dylan Thurston [33], Sylvain Poirier [34] and others'. If k = 0, then Z(0, M) is the Kontsevich configuration space invariant ZKKT (M),which is a universal invariant for Z-spheres according to a theorem of Kuperberg and Thurston [9; 13],and which was completed to a universal finite type invariant for Q-spheres by Delphine Moussard [21]. The proof of this theorem is sketched in Section 5.
Under its assumptions, let ®0 be a homogeneous propagating form of (C2(M), t), let i be
1
the involution of C2(M) that permutes two elements in M2\diagonal,set © = — (©„ -i*(®0)),
and set ©(i) = © for any i.
Let Aut(r) be the set of automorphisms of r, which is the set of permutations of the half-edges that map a pair of half-edges of an edge to another such and a triple of half-edges that contain a vertex to another such, and that map half-edges of univalent vertices on a component Kj to half-edges of univalent vertices on Kj so that the cyclic order among such vertices is preserved. Set
p = (3n - #E(r))! Pr (3n)!2#E(r) .
* After work of many people including Witten [35], Guadagnini, Martellini and Mintchev [36], Kontsevich [37; 38], Bott and Taubes [39], Bar-Natan [40], Axelrod and Singer [41; 42].
Then Zre^ (C)Pr I(r,(®(i))ie3n )[r] reads
1
£ #Atir> IM«»"* >F1
r unnumbered, unoriented ■"■■ii-U.tVl /
where the latter sum runs over the degree n Jacobi diagrams on C without looped edges.
1
Indeed, for a numbered graph r, there are — ways of renumbering it, and #Aut(r) of them
Pr
will produce the same numbered graph. 4.4. On the universality proofs
Theorem 6. Let y, z e N. Recall y = {1,2,...,y}. Set (z + y) = {y + 1,y + 2,...,y + z}. Let M be an asymptotically standard q-homology M3. Let L be a link in M. Let (Bb) be a collection of pairwise disjoint balls in M such that every Bb intersects L as a ball of a crossing change that contains a positive crossing cb, and let L((Bb)bey) be the link obtained by changing the positive crossings cb to negative crossings. Let (Aa )ae(z+y) be a collection of pairwise
disjoint rational homology handlebodies in M \ (L Bb). Let (A'a /Aa) be rational LP surgeries in M . Set X = [M, L;(A'a / Aa)ae(z+y),(Bb, cb)b^y] and define Zn(X) as the sum over all
subsets I of y+z of the terms (-1)# 1 Zn (L((Bb )b^Iny ), M(( A'a / )aElnil+y})). If 2n < 2y + z, then Zn (X) vanishes.
Sketch of proof. As in [13], one can use (generalized) propagators for the M(( A'a / Aa n(z+y)) that coincide for different I wherever it makes sense (for example, for configurations that do not involve points in surgered pieces Aa). See also [9]. Then contributions to the alternate sum of the integrals over parts that do not involve at least one point in an Aa or in an Aa, for all a cancel. Assume that every crossing change is performed by moving only one strand. Again, contributions to the alternate sum of the integrals that do not involve at least one point on a moving strand cancel. Furthermore, if the moving strand of cb is moved very slightly, and if no other vertex is constrained to lie on the other strand in the ball of the crossing change, then the alternate sum is close to zero. Thus in order to produce a contribution to the alternate sum, a graph must have at least (2y + z) vertices. See [32] or [22. Section 5.4], and [13. Section 3] for more details. □
This implies that Zun is of degree at most n for links in M3, and that Zn is of degree at most 2n for Z-spheres or Q-spheres.
Now, under the hypotheses of Theorem 6, assume that Aa is the standard genus 3 handlebody with three handles with meridians m(a) and longitudes such that (m(ia},£ J>dA = Siy. See Aa as a thickening of the trivalent graph on Fig. 2.
Also assume that Aa is an integer homolo-gy handlebody. In Aa udA (-A'a), there is a surface S, such that d(S, n Aa ) = mJa). Assume that
j J a J
(S1,S2,S3V , „ =1. (For example, choose A'
x 1 2 3 Aa udAa Aa > a
such that Aa usAa A a) = like in the case of
the Matveev Borromean surgery of [43].) Assume mo that the l\a) bound surfaces D\a) in M.
iii'S J J
Fig. 2
Assume that the collection of surfaces ja)}ae(z+y),je3 reads {DpA}pep U {D^}pe£ so that for
any q e P, for 5e 2.
if d = n(a(q,5))
lf Dq,5 Dj(q,&) ,
the Interior of Dq 5 Intersects
q,5
L — U
ae(z+ y) -\(a(q,3-5))
Aa - - (a) Dj
a je3,Dja *Dq,5 1
(a)
— —
bey
(B)
only in Aa(q,3-5) — IE?)
Note that {Dqfi, j^»)M = lk(dDqV dDj.
Example 3. Note that these assumptions are realised in the following case. Start with an embedding of a Jacobi diagram r whose univalent vertices belong to chords (plain edges between two univalent vertices) on — ki=i Sj in M. Assume that the trivalent vertices of r are labeled in (z + y), and assume that its chords are labeled in y. Apply the following operations replace
edges >-< without univalent vertices by Mj^K , replace a chord •—' labeled by b by a
crossing change cb j j V--' in a ball Bb that is a neighborhood of the plain edge. Thicken
the trivalent graph associated to the trivalent vertex labeled by a, and call it Aa. Then the
surfaces Djo) are the disks bounded by the small loops of
Conversely, under the assumptions before the example, define the following vertex-oriented Jacobi diagram r([M,L;(A'a/Aj^+^XB,cXj) on — k=jSj, with:
• two univalent vertices joined by a chord for each crossing change ball Bb at the corresponding places on —k=i Sj (in L'i(Bb)),
• one trivalent vertex for each Aa, where the three adjacent half-edges of the vertex correspond to the three D(a\ with the fixed cyclic order,
such that any pair of half-edges corresponding to some Dp i and its friend Dp 2 forms an edge between two trivalent vertices.
Theorem 7. Under the assumptions above, let X = [M, L;(A'a / Aa)a^(¿+y), (Bb, Cb ] 2n = 2y + z,
Zn (X) =
nik(3D,
V PeP
p,i, dDp,2)
[r(Z)]modir
or in
At № )
j=j
(1T)
Sketch of proof. When z = 0, the proof of Theorem 6 can be pushed further in order to prove the result like in [32] or [22. Section 5.4]. In general, when y = 0, it is a consequence of the main theorem in [13] (Theorem 2.4). The general result can be obtained by mixing the arguments of [13. Section 3] with the arguments of the link case. □
This theorem is the key to proving the universality of Zu among Vassiliev invariants for links in M3 and to proving the universality of Z among finite type invariants of Z-spheres. This universality implies that all finite type invariants factor through Z.
Remark 3. Theorem 7 with ZLMO instead of Z is proved in [20], when y = 0, when the (Aa /Aa) are Matveev's Borromean surgeries and when the D(a) are disks such that lk(3D vdDp2) = i. Then the main theorem of [44] implies Theorem 7 with ZLMO instead of Z, when y = 0 and when the Aa and the Aa are integral homology handlebodies.
5. Compactifications, anomalies, proofs and questions
In this section, we state Theorem 8. This is another version of Theorem 5, which leads to a definition of Z involving algebraic intersections rather than integrals in Subsection 5.3. It is based on the concept of straight links introduced in Subsection 5.2.
This section also contains sketches of proofs of Theorems 5 and 8. We begin with the introduction of appropriate compactifications of configuration spaces to justify the convergence of our integrals stated in Proposition 6.
5.1. Compactifications of configuration spaces
Let N be a finite set. See the elements of MN as maps m: N ^ M.
For a non-empty I ç N, let EI be the set of maps that map I to œ. For I ç N such that # I > 2, let AI be the set of maps that map I to a single element of M. When I is a finite set, and when V is a vector space of positive dimension, SI(V) denotes the space of injective maps from I to V up to translation and dilation. When # I > 2, SI(V) embeds in the compact space SI (V) of non-constant maps from I to V up to translation and dilation.
Lemma 23. The fiber of the unit normal bundle to AI in MN over a configuration m is Si (Tm(I) M).
Proof. Exercise. □
Let CN(M) denote the space of injective maps from N to M. Define a compactification CN (M) of CN (M) by generalizing the previous construction of C2(M) = C2(M) as follows.
Start with MN. Blow up EN, which is the point m = œN such that m_1(œ) = N. Then for k = #N,#N _1,...,3,2, in this decreasing order, successively blow up the (closures of the preimages under the composition of the previous blow-down maps of the) AI such that # I = k (choosing an arbitrary order among them) and, next, the (closures of the preimages under the composition of the previous blow-down maps of the) Ej such that # J = k _1 (again, choosing an arbitrary order among them).
Lemma 24. The successive manifolds that are blown-up in the above process are smooth and transverse to the boundaries. The manifold CN (M) is a smooth compact (3#N)-manifold independent of the possible order choices in the process. For i, j e N, i ^ j, the map
Pj : CN(M) ^ C2(M)
m ^ (m(i), m( j))
smoothly extends to CN (M).
Sketch of proof. A configuration m0 of MN induces the following partition C(m0) of
N = m0-1(œ^ ]J m0-1(x).
xeM omo(N)
Pick disjoint neighborhoods Vx in M of the points x of m0(N) that are furthermore in M for x in M and that are identified with balls of M3 by Cœ-charts. Consider the neighborhood
n, V0 (x) of m„ in MN. The first blow-ups that transformed this neighborhood are:
xem0 (N) x 0 1 "
• the blow-up of E if m,-1(œ) ^ 0, which changed (a smaller neighborhood of œm- (œ)
m0 (œ) 0
in) Vœm01(œ) to [0, Sœ [xS3#m01(œ)_4;
• and the blow-ups of the A , for the x e M such that # m01 (x) > 2, which changed
m)(x) 0
(a smaller neighborhood of xm°1(x) in) Vxm°1(x) to [0,sx[xF(Um0_1(x)), where Ux c Vx and F(Uxm°1(x)) fibers over U , and the fiber over y e Ux is S . (TyM).
x u x m_ (x) y
When considering how the next blow-ups affect the preimage of a neighborhood of m0, we can restrict to our new factors.
First consider a factor [0,sx[xF(U"m0 (x)). Picking i e m0_1(x) and fixing a Riemannian
structure on TUx identifies S < (T,M) with the space of maps c : mnx(x) ^ T M such that
x m-1 (x) y 0 y
c(i) = 0 and V J c(j) ||2= 1. Then (X, c) is identified with y | Xc in Vrm-V) (where Vx is
¿-^jemQ (x) 11 J 11 ' x x
identified with an open subset of M3), for X ^ 0. Now, [0,sx[xF(Um (x)) must be blown-up along its intersections with the preimage closures of the AI such that # I > 2, I c m01 (x) and I is maximal. These intersections respect the product structure by [0, sx [ and the fibration over Ux so that we only need to understand the blow-ups of the intersections of the AI with a fiber of
F(Um (x)). These are nothing but configurations in a ball of M3, and we can iterate our process.
Now consider the possible factor [0, sO[x53#m° (oo) 1 and blow up its intersections with the
preimage closures of the Ej for J c m01(o) maximal and with the preimage closures of the
AI with I c m01(o) in an order compatible with the algorithm. Here, S3#m° (o) 1 is the unit
sphere of (MoO )m A point d e (MOO )m0 ° is in the preimage closure of EJ under the previous blow-up if d(j) = 0. In particular, the EJ and the AI again read as products by [0,so[,
and we study what happens near a given d of S"3#m° (o)°1. For such a d, we proceed as before if d_1(0) = 0. Otherwise the factor of d_1(0) must be treated differently, namely by blowing up
0d°1(0) in 53#m°1(o)°1. Then iterate.
This produces a compact manifold CN(M) with boundary and ridges, which is finally independent of the order of the blow-ups (when this order is compatible with the algorithm), since it is locally independent. The interior of CN(M) is CN(M). Since the blow-ups separate all the pairs of points at some scale, pe naturally extends there. The introduced local coordinates show that the extension is smooth. See [31. Section 3] for more details.
Lemma 25. The closure of C(L; r) in CV(r)(M) is a smooth compact submanifold of CV(r)(M), which is denoted by C(L;r). Proof. Exercise. □
Proposition 6 is a consequence of Lemmas 24 and 25.
5.2. Straight links
A one-chain c of S2 is algebraically trivial if for any two points x and y outside its support, the algebraic intersection of an arc from x to y transverse to c with c is zero, or equivalently if the integral of any one form of S2 along c is zero.
Let (M,t) be an asymptotic rational homology M3. Say that Kj is straight with respect to t if the curve pT (U+Kj) of S2 is algebraically trivial (recall the notation from Proposition 2 and Subsection 4.2). A link is straight with respect to x if all its components are. If Kj is straight, then pz(dC(Kj; rrySj) ) is algebraically trivial.
Lemma 26. Recall Cj = C(K ', ), (cC(M). If pz(8Cj) is algebraically trivial, then for any propagating chain V of (C2(M).x) transverse to Cj and for any propagating form <ap of (C2(M), t) ,
i = (Cj,V> M)= /e(Kj,t)
j
where Ie(K},t) is defined in Lemma 22. In particular, Ie (K},t) e Q and Ie (Kj,t) e Z when M is an integer homology 3-sphere. Proof. Exercise. Recall Lemmas 9 and 21. □
Proposition 7. Let M be an asymptotically standard Q-homology M3. For any parallel K of a knot K in M ,there exists an asymptotically standard parallelization t homotopic to t, such that K is straight with respect to t ,and Ie (K}, t) = lk(K, K) or Ie (Kj, x) = lk(K,K) +1.
For any embedding K: S1 ^ M that is straight with respect to t , Ie (K, t) is the linking number of K and a parallel of K.
Sketch of proof. For any knot embedding K, there is an asymptotically standard paralleli-zation T homotopic to t such that pT(U+K) is one point. Thus K is straight with respect to (M, t) . Then t induces a parallelization of K, and /0 (K, T) is the linking number of K with the parallel induced by T. (Exercise).
In general, for two homotopic asymptotically standard parallelizations t and T such that K is straight with respect to t and T , I0 (K,t) - I0 (K,T) is an even integer (exercise) so that I0 (K,t) is always the linking number of K with a parallel of K.
In R3 equipped with Ts, any link is represented by an embedding L that sits in a horizontal plane except when it crosses under, so that the non-horizontal arcs crossing under are in vertical planes. Then the non-horizontal arcs have an algebraically trivial contribution to pz(U+K ), while the horizontal contribution can be changed by adding kinks or so that L is straight with respect to Ts. In this case I0(K},ts) is the writhe of Kj, which is the number of positive self-crossings of Kj minus the number of negative self-crossings of Kj. In particular, up to isotopy of L, /0 (Kj, ts ) can be assumed to be ±1. (Exercise).
Similarly, for any number i that is congruent mod 2Z to I0 (K, t) there exists an embedding K' isotopic to K and straight such that I0(K',t) = i. (Exercise).
5.3. Rationality of Z
Let us state another version of Theorem 5 using straight links instead of homogeneous propagating forms. Recall pr = ——# .
r (3n)'2#1 (r)
Theorem 8. Let (M,t) be an asymptotic rational homology R3. Let L:]_[k S^ ^M be a straight embedding with respect to t. For any i e 3n, let ®(i) be a propagating form of (C2(M), t) . Then
X Pr/(r>(0),eü[r] e V'n I JJSJ
TeVen (C) V J-1 ,
is independent of the chosen ®(i). It is denoted by Zsn(L,M,t). In particular, with the notation of Theorem 5, Z"n (L,M, t) = Zn (L,M,t).
This version of Theorem 5 allows us to replace the configuration space integrals by algebraic intersections in configuration spaces, and thus to prove the rationality of Z for straight links as follows.
For any i e 3n, let V(i) be a propagating chain of (C2(M),t). Say that a family (V(i))ie3n is in general 3n position with respect to L if for any re Ven (C), the p-(V(jE(e))) are pairwise transverse chains in C(L; r). In this case, define I (r, o(r),(V (i))ie3n) as the algebraic intersection in (C(L;r),o(r)) of the codimension 2 rational chains p-\V(jE(e))). If the ®(i) are propagating forms of (C2(M), t) Poincaré dual to the V(i) and supported in sufficiently small neighborhoods of the V(i), then /(r, o(Y),(V (i))^ ) = I(r, ^»(O),^) for any reP; (C), and I(r, o(r)5(ffl(/))ie3S) is rational, in this case.
5.4. On the anomalies
The constants a = (an)neN and p = (Pn)neN of Theorem 5 are called anomalies. The anomaly P is the opposite of the constant £ defined in [31. Section 1.6], p2n =0 for any integer n, and
Pi = according to [31. Proposition 2.45]. The computation of Pj can also be deduced
from Corollary 1.
We define a below. Let v e S2. Let Cv denote the linear map
Cv : R ^ R3 1 ^ v.
Let r be a numbered Jacobi diagram on M. Define C(£v;r) like in Subsection 4.1 where the line Cv of M3 replaces the link L of M. Let Q(v; r) be the quotient of C(£o;T) by the translations parallel to Cv and by the dilations. Then the map p associated to an edge e of
r maps a configuration to the direction of the vector from its origin to its end in S2. It factors through Q(v; r), which has two dimensions less. Now, define Q(r) as the total space of the fibration over S2 whose fiber over v is Q(v; r). The configuration space Q(r) carries a natural smooth structure, it can be compactified as before, and it can be oriented as follows, when a vertex-orientation o(r) is given. Orient C(Lv; r) as before, orient Q(v; r) so that C(£v;T) is locally homeomorphic to the oriented product (translation vector z in Mv, ratio of homothety A e]0, <x>[) xQ(v; r) and orient Q(r) with the (base(=S2) © fiber) convention. (This can be summarized by saying that the S2-coordinates replace (z, A).)
Proposition 8. For i e 3n,let ©(i, S2) be a two-form of S2 such that j 2©(i, S2) = 1. Define I(r, o(r), ©(i, S2)) as S
ix A p*,2(®(jE(e),S2)).
Let Dcn (M) denote the set of connected numbered diagrams on M with at least one univalent vertex, without looped edges. Define the element 2an of A(M) as
I (3n(r))! I(r,o(r),©(i,S2))[r, o(r)].
reDn (M) (3n)!2
1 r * 1
Then a„ does not depend on the chosen ©(z, 52), a, = -|_C j and a2k = 0 for all k e N. The series a = I ^an is called the Bott and Taubes anomaly.
Proof. The independence of the choices of the ©(i,S2) will be a consequence of Lemma 27 below. Let us prove that a2k = 0 for all k e N . Let r be a numbered graph and let r be obtained from r by reversing the orientations of the (# E) edges of r. Consider the map r from Q(r) to Q(r) that composes a configuration by the multiplication by (-1) in M3. It sends a configuration over v e S2 to a configuration over (-v), and it is therefore a fibered map over the orientation-reversing antipode of S2. Equip r and r with the same vertex-orientation. Then our map r is orientation-preserving if and only if # T(r) + 1 + # E(r) is even. Furthermore for all the edges e of r, p ° r = p 2, then since # E = n + # T,
I(F, o(r), ©(i, S2)) = (-1)n+1l(r, o(r), ©(i, S2)). □
It is known that a3 = 0 and a5 = 0 [34]. Furthermore, according to [45], a2n+1 is a combination of diagrams with two univalent vertices, and Zu (S3, L) is obtained from the Kontsevich integral by inserting d times the plain part of 2a on each degree d connected component of a diagram.
5.5. The dependence on the forms in the invariance proofs
The variation of I(r,o(r),(©(j))je3n) when some ©(i = jE(f e E(r))) is changed to ©(i) + dn
for a one-form n on C2(M) reads
f ( ^
Jr(fr) P(dn) a a peje(e»> -
(L- ' y ec(E(T)\{ f}) ^
where C(L;T) is equipped with the orientation induced by o(r). According to the Stokes theorem, it reads
( \
Pf (n> A A PijE
ldC(L\T)
€<E(r)l{ f})
where the integral along d(C(L;r),o(r)) is actually the integral along the codimension one faces of C(L;T), which are considered as open. Such a codimension one face only involves one blow-up.
For any non-empty subset B of V(r), the codimension one face associated to the blow-up of EB in MV(r) is denoted by F(r, a>, B), it lies in the preimage of B x Mv(r)B, in C(L;r).
The other codimension one faces are associated to the blow-ups of the AB in MV(r), for subsets B of V(r) of cardinality at least 2. The face of C(L;T) associated to AB is denoted by F(r;B). Let beB. Assume that b e U(r) if U(r) n B * 0. The image of F(r;B) in MV(r) is in the set of maps m of AB that define an injection from (V(r) IB) u {b e B} to M, which factors through an injection isotopic to the restriction of ir on C/(r) n (((r) I B) u {b}) .This set of maps C(V(r)IB)ij{b)(M,ir) is a submanifold of C-^^^^CM). Thus, F(r;B) is a bundle over
C(v(T)\B)u{b)(M, V).
When B has no univalent vertices, the fiber over a map m is the space SB(Tmib)) of injective maps from B to Tm(b) up to translations and dilations.
When B contains univalent vertices of a component Kj, the fiber over m is the submanifold SB (Tm(b) M, r) of SB (Tm(b) M), made of the configurations that map the univalent vertices of B to a line of Tm{b)M directed by U+ Kj at m(b), in an order prescribed by r. If B does not contain all the univalent vertices of r on Si, this order is unique. Otherwise, F(r, B) has #(B n U(r)) connected components corresponding to the total orders that induce the cyclic order of B n U(r).
When B is a subset of the set of vertices V(r) of a numbered graph r, E(rB) denotes the set of edges of r between two elements of B (edges of r are plain), and rB is the subgraph of r made of the vertices of B and the edges of E(rB).
Lemma 27. Let (M, t) be an asymptotic rational homology M3. Let C = JJ^Sy.
For i e 3n, let ®(i) be a closed 2-form on [0,i] x C2(M) whose restriction to {t} x C2(M) is denoted by ra(i, t), for any t e [0,i]. Assume that for t e [0,i], ra(i,t) restricts to (dC2(M) I UBM) as p*(®(i,t)(S2)), for some two-form o>(i,t)(S2) of S2 such that j^2©(i, t)(S2) = i. Set
Zn(t) = Zr^(C)Pr^(r,(®(i,t))i63n)[r] in At(Uk=iSi). Then Zn(i) - Zn(0) = ^I(r,B) where the sum runs over the set (r,B)
{(r,B);r e T>e(C),B c V(r),#B > 2; rB is a connected component of r}
and
I(r, B) = Prj[0,i]xF( r, B) eeAp>( 1e (e)))[r].
Under the assumptions of Theorem 5 (where the ®(i) are homogeneous) or Theorem 8 (where L is straight with respect to t), when (M, L, t) is fixed, Zn (L, M, t) is independent of the chosen ®(i).
In particular, when k = 0, Z(M, t) coincides with the Kontsevich configuration space integral invariant described in [31].
Furthermore,the an of Proposition 8 are also independent of the forms a>(i,S2).
Sketch of proof. According to the Stokes theorem, for any re£>en(C),
I(r,(ffl(i,l))ie3n) - I(r,(<o(i,0)U,) = ZLf A P<®<h(e)))
F [0,1 eeE(T)
where the sum runs over the codimension one faces F of C(L;T). Below, we sketch the proof that the only contributing faces are the faces F(r, B) such that # B > 2 and TB is a connected component of r, or equivalently, that the other faces do not contribute.
Like in [31. Lemma 2.17] faces F(T,x>,B) do not contribute. When the product of all the pe factors through a quotient of [0,1] x F(r, B) of smaller dimension, the face F(r, B) does not contribute. This allows us to get rid of
• the faces F(r, B) such that B is not a pair of univalent vertices of r, and TB is not connected (see [31. Lemma 2.18]);
• the faces F(r, B) such that # B > 3 where TB has a univalent vertex that was trivalent in r (see [31. Lemma 2.19]).
We also have faces that cancel each other, for graphs that are identical outside their TB part:
• the faces F(r, B) (that are not already listed) such that TB has at least a bivalent vertex cancel (mostly by pairs) by the parallelogram identification (see [31. Lemma 2.20]);
• the faces F(r, B) where TB is an edge between two trivalent vertices cancel by triples, thanks to the Jacobi (or IHX) relation (see [31. Lemma 2.21]);
• similarly, two faces where B is made of two (necessarily consecutive in C) univalent vertices of r cancel (3n - # F(r)) faces F(r',B') where ris an edge between a univalent vertex of r and a trivalent vertex of r, thanks to the STU relation.
Thus, we are left with the faces F(r, B) such that TB is a (plain) connected component of r, and we get the wanted formula for (Zn(1) - Zn(0)).
In the anomaly case, the same analysis of faces leaves no contributing faces, so that the an are independent of the forms ©(i, S2) in Proposition 8.
Back to the behaviour of Z(L, M, t) under the assumptions of Theorem 5 or Theorem 8, assume that (M, L, t) is fixed and apply the formula of the lemma to compute the variation of Zn (L, M, t) when some propagating chain ©(i,0) of (C2(M), t) is changed to some other propagating chain ©(¿,1) = ©(i, 0) + dn. According to Lemma 9, under our assumptions, n can be chosen so that n = p*(ns2) on dC2(M) and n 2 =0 if ©(i,0) and ©(i,1) are homogeneous.
Define ©(i) = ©(i,0) + d(tn) on [0,1] x C2(M) (t e [0,1]), and extend the other ©(j) trivially.
Then (Zn(1) - Zn(0)) vanishes if ©(i,0) and ©(i,1) are homogeneous, as all the involved I(r, B) do, so that Zn (L, M, t) is independent from the chosen homogeneous propagating forms ©(i) of C2(M, t) in Theorem 5. Now, assume that L is straight.
When i £ JE(E(r)) , the integrand of I(r, B) factors through the natural projection of [0,1] x F(r, B) onto F(r, B) , so that I(r, B) = 0. Assume i = jE(ei e E(r)) , then I(r, B) equals
PrL^A (d(tn)) A A p(®(jE(e)).
e^E(T)\ei
The form ^eeE(r p*(©( jE(e)) pulls back through [0,1] x F(rB, B), and through F(rB, B) when ei £ E(rB), so that, for dimension reasons, I(r,B) vanishes unless ei e E(rB). Therefore, we assume ei e E(rB).
When B contains no univalent vertices, I(r, B) factors through the integral along
[0^] X *U m(b)eM SB (Tm(b)M) of
p* (d(tn)) a A pK^He(e))). Here the parallelization t identifies the bundle u
n(bhM SB (Tmb)M) M x SB (m3 )
integrand factors through the projection of [0,1] x M x SB(M3) onto [0,1] x SB(M3) whose dimension is smaller (by 3). In particular, I(r, B) = 0 in this case, the independence of the choice of the ©(i) is proved when k = 0 (when the link is empty), and Z(M, t) coincides with the Kontsevich configuration space integral invariant described in [31].
Let us now study the sum of the I(r, B), where (r \ rB) is a fixed labeled graph and TB is a fixed numbered connected diagram with at least one univalent vertex on S1.
This sum factors through the integral along [0,1] x um(b)eK SB(Tm(b)M, r) of
p" (d(tn)) a a p*e(w(jE(e))).
e^E(rB )\ej
At a collapse, the univalent vertices of rB are equipped with a linear order, which makes rB a numbered graph rB on M. The corresponding connected component of [0,1]x
^m(b)eK ■ SB (Tm(b)M, r) reads [0,1] x ^ , +K Q(pT (x); rB ) Q(v; r B)
section 5.4). This allows us to see the contribution of such a connected component as the integral of a one-form (defined by partial integrations) over pT (U + Kj). Such an integral is zero when Kj is straight. □
Now, Theorem 8 is a corollary of Theorem 5 (which is not yet completely proved).
5.6. The dependence on the parallelizations in the invariance proofs
Recall that Vtn (C) splits according to the number of connected components without univalent vertices of the graphs. Then it is easy to observe that
Z(L, MM, t) = YsZn (L, MM, t) = Zu (L, MM, t)Z(M; t)
neN
where Z" is obtained from Z by sending the graphs with components that have no univalent vertices to 0, and Z(M;t) = Z(0,M,t). According to [31, Theorem 1.9], Z(M) = Z(M;x)exp ( -1 p4(x)p |
is a topological invariant of M. Here, we will now focus on Zu(L, M, t) , and define it with a given homogeneous propagating form, © = ©(i) for all i, so that Zu(L, M, t) is an invariant of the diffeomorphism class of (L, M, t). We study its variation under a continuous deformation of t and we prove the following lemma.
Lemma 28. Let (T(O)ie[01] define a smooth homotopy of asymptotically standard paral-
d ~
lelizations of M. Then —Zu (L, M, T(t)) is equal to
dt
f k d ^
Zu (L, M, x(t)).
Ie (Kj, T(t))a # j
V j=i dt
Proof. Set Zn(t) = Z"(L, M, x(t)), observe that Zn (which is valued in a finite-dimensional vector space) is differentiable thanks to the expression of Zn (t) - Zn(0) in Lemma 27 (any
function J for a smooth compact manifold C and a smooth form ® on [0,1] x C is differentiable with respect to t). Now, the forms associated to edges of rB do not depend on the configuration of (y(r)\B). They will be integrated along [0,1] x (^m(b)eKj SB(Tm{b)M, rB)) while the other ones will be integrated along C(L; r I r B ) at we [0,1]. Therefore, the global variation (Z(t) - Z(0)) reads
k ( \ k ft
Z(u)du
EJ0 E Mb№B]#j
1-1 ^rB<E»c (M) )
where (R) = uneN Vcn (R) and IB(") is the integral along {c e um(b)eK.SB(Tm(b)M, rB)} of
(A e,E(TB p (wS2>)(m, c). Define I(rB, Kj )(t) as the integral along
{(u c); u e [0,t], c e Um(b) eKj SB (Tm(b)'M' , ^
» d d of A P*e2)(u,c), so that IB(u) = — I(rB,K.)(u)du. Therefore, —Z(t) reads
eErB) 5 d" dt
* id ) X X PrB KjXt)[TB]#j z(t)
j=l yTb^Vc (R) dt d
and we are left with the computation of —I(rB, Kj )(t).
dt J
The restriction of pT() from [0,1] x U+Kj to S2 induces a map
Sb (Tm(b)M, rB ) ^ Q(rB )
for any rB,
l(rB'K)(t) = L(P ) A p>52>.
Integrating ^ eETB pe (®52)[rB ] along the fiber in Q(rB) yields a two-form on S2, which is homogeneous, because everything is. Thus this form reads 2a(rB)©2[rB] where a(rB) e R, and where ^^ (R)3rB a(r B )[r b ] = a. Therefore
l(rB, K m = 2PrB a(r b
Since — f + p*()(w 2) = 1 — I(K:,x(t)), we conclude easily. □
dt }[Q,t]xu+K/ t( ■ ^ s2 2 dt
Then the derivative of
fj exp(-Ie (Kj, T(t))a) #j Zu (L, M, x(t))
j=i
vanishes so that this expression does not change when t smoothly varies. 5.7. End of the proof of Theorem 5
Thanks to [31. Theorem 1.9], in order to conclude the (sketch of) proof of Theorem 5, we are left with the proof that
17 ( exp(-1 (Kj, T)a) # j) Zu (L, M, t)
j=1
does not depend on the homotopy class of t.
When t changes in a ball that does not meet the link, the forms can be changed only in the neighborhoods of the unit tangent bundle to this ball. Using Lemma 27 again, the variation will be seen on faces F(r,B), where TB has at least one univalent vertex, and where the forms associated to the edges of TB do not depend on the parameter in [0,1] so that their product vanishes. In particular,
17 ( exp(-1 (Kj, T)a) # j) Zu (L, M, t)
j=1
is invariant under the natural action of n3(SO(3)) on the homotopy classes of parallelizations.
We now examine the effect of the twist of the parallelization by a map g : (BM, 1) ^ (SO(3), 1). Without loss, assume that pT (U+Kj) = v for some v of S2 and that g maps Kj to rotations with axis v. We want to compute
Zu(L, M, t o yR(g)) - zu(L, M, t). Identify UBM with BM x S2 via t. There exists a form © on [0,1] x BM x S2 that reads p*(© 2) on d([0,1] x BM x S2) I (1 x BM x S2) and that reads p* (g)(©2) on 1 x BM x S2. Extend this form to a form Q on [0,1] x C2(M), that restricts
to 0 xdC2(M) as P*(©S2 ) , and to 1 xdC2(M) as P*=vr (g)(©S2), where Pt°vr (g) = Pt 0 Vr ^^ on
B„ x S2
M
so that p*o^(g)(©2) = yR(g 4)* (p*(©2)), there. Let Venu(C) denote the set of diagrams
of Ven (C) without components without univalent vertices. Define Zn(t) e Vn(J J^1) by
Zn (t)= X Prl(F,(ntxC
(M) ^ieSn
r^ve"(c)
For rB eV c(R), define I(rB, K. ,Q)(t) as the integral along
{(U c); u e [0,i], c e Vm(b)eKj SB(Tm(b)M, rB)}
of A e.E(rB Pe ].
d
Set pj(t) = XrB,EDcKj,Q)(t) and y.(t) = — P.(t). Thanks to Lemma 27, like in the
proof of Lemma 28, Z(t) is differentiable, and Z'(t) = ^Y/(t)#^Z(t).
By induction on the degree, it is easy to see that this equation determines Z(t) as a function of the P..(t) and Z(0) whose degree 0 part is 1, and that Z(t) = H*=1 exp(p.(t))#. Z(0).
Extend Q over [0,2] x C2(M) so that its restriction to [1,2] x BM x S2 is obtained by applying (yR(g-1)) to the Q translated, and extend all the introduced maps, then y■ (t + 1) = y.(t) because everything is carried by (yR(g-1)) . In particular p.(2) = 2p..(1). Now, Z(2) = Zu(L,M,t o yR(g)2) is equal to
fl exp((I0(Kj, t o (g)2) - I0 (Kj, x))a) #. Z(0),
j=1
where Z(0) = Zu (L, M, t), since g2 is homotopic to the trivial map outside a ball (see Lemma 29, 2). By induction on the degree of diagrams, this shows
P.(2) = (I0(Ky, t o (g)2) - I0(Ky, T))a.
Conclude by observing that under our assumptions, where I0 (K., t o (g)1) is the linking number of K ■ and its parallel induced by t o yr (g)1, I0 (K., t o yR (g)2) -10 (K., t) is equal to 2(Ie(K.,t o yR(g)) -10(K.,t)). This finishes the (sketch of) proof of Theorem 5 in general.
5.8. Some open questions
1. A Vassiliev invariant is odd if it distinguishes some knot from the same knot with the opposite orientation. Are there odd Vassiliev invariants?
2. More generally, do Vassiliev invariants distinguish knots in S3?
3. According to a theorem of Bar-Natan and Lawrence [46], the LMO invariant fails to distinguish rational homology 3-spheres with isomorphic H1, so that, according to a Moussard theorem [21], rational finite type invariants fail to distinguish Q-spheres. Do finite type invariants distinguish Z-spheres?
4. Find relationships between Z or other finite type invariants and Heegaard Floer homologies. See [6] to get propagators associated to Heegaard diagrams. Also see related work by Shimizu and Watanabe [47; 48].
5. Compare Z with the LMO invariant ZLMO.
6. Compute the anomalies a and p.
7. Find surgery formulae for Z.
8. Kricker defined a lift ZK of the Kontsevich integral ZK (or the LMO invariant) for null-homologous knots in Q-spheres [49; 50]. The Kricker lift is valued in a space A that is mapped to Vn(Si) by a map H, which allows one to recover ZK from ZK. The space A is a space of trivalent diagrams whose edges are decorated by rational functions whose denominators divide the Alexander polynomial. Compare the Kricker lift ZK with the equivariant configuration space invariant Zc of [51] valued in the same diagram space A. See [52] for alternative definitions and further properties of Zc.
9. Is Z obtained from Zc in the same way as ZK is obtained from ZK?
6. More on parallelizations of 3-manifolds and Pontrjagin classes
In order to make the definition of ® complete, we give a detailed self-contained presentation of p1(t). In this section, M is a smooth oriented connected 3-manifold with possible boundary.
6.1. [(M,dM),(SO(3),1)] is an abelian group.
Again, see S3 as B3/dB3 and see B3 as ([0,2n] x S2) / (0~{0} x S2). Recall that p : B3 ^ SO(3) maps (9 e [0,2n],v e S2) to the rotation p(0,v) with axis directed by v and with angle 0.
Also recall that the group structure of [(M, dM),(SO(3),1)] is induced by the multiplication of maps, using the multiplication of SO(3). Any g e C0 ((M, dM),(SO(3),1)) induces a map
Hx(g) : H(M,dM) ^ iH1(50(3),1) = Z
y 2Z
where coefficients are in Z unless otherwise mentioned, so that H4(g) = H4(g; Z) and H (M, dM) = H1 (M, dM; Z). Since
( Z ^ Z
H I M, dM;— I = H1(M, dM) / 2H1(M, dM) = H1(M, dM) ®Z —,
y 2ZJ 2Z
Hom^H1(M,dM),Zjj is isomorphic to
' ' ^ ^ = H1 f M,dM; —
1 2Z
Homl H I M,dM;— I,— 1 11 2Z J 2Z
and the image of H4(g) under the above isomorphisms is denoted by H4(g;Z / 2Z). (Formally, this H4(g;Z/21) denotes the image of the generator of H4(SO(3),1;Z/2Z) = Z/2Z under H'(g;Z/2Z) in Hl(M,dM;Z/^).)
Lemma 29. Let M be an oriented connected 3-manifold with possible boundary. Recall that pM(B3) e C0 ((M, 5M),(SO(3),1)) is a map that coincides with p on a ball B3 embedded in M and that maps the complement of B3 to the unit of SO(3).
1. Any homotopy class of a map g from (M,dM) to (SO(3),1), such that Hl(g;Z/2Z) is trivial,belongs to the subgroup <[pM(B3)]) of [(M,dM),(SO(3),1)] generated by [pM(B3)].
2. For any [g] e [(M,5M),(SO(3),1)], [g]2 e<[pM(B3)]).
3. The group [(M,dM),(SO(3),1)] is abelian.
Proof. Let g e C0 ((M, dM),(SO(3),1)). Assume that H4(g;Z/21) is trivial. Choose a cell decomposition of M with respect to its boundary, with only one three-cell, no zero-cell if dM = 0, one zero-cell if dM = 0, one-cells, and two-cells. Then after a homotopy relative to dM, we may assume that g maps the one-skeleton of M to 1. Next, since n2(SO(3)) = 0, we may assume that g maps the two-skeleton of M to 1, and therefore that g maps the exterior of some 3-ball to 1. Now g becomes a map from B3 / dB3 = S3 to SO(3), and its homotopy class is k[p] in n3(SO(3)) = Z[p]. Therefore g is homotopic to pM(B3)k. This proves the first assertion.
Since H4(g2;Z / 2Z) = 2H4(g;Z / 2Z) is trivial, the second assertion follows.
For the third assertion, first note that [pM(B3)] belongs to the center of [(M,dM),(SO(3),1)] because it can be supported in a small ball disjoint from the support (preimage of SO(3) \ {1}) of a representative of any other element. Therefore, according to the second assertion any square will be in the center. Furthermore, since any commutator induces the trivial map on n4(M), any commutator is in <[pM(B3)]). In particular, if f and g are elements of [(M,dM),(SO(3),1)], (gf)2 = (fg)2 = (f f2g2f )(f {g fg) where the first factor equals f2g2 = g2f2. Exchanging f and g yields f ~lg~lfg = g~lf ~lgf. Then the commutator, which is a power of [pM(B3)], has a vanishing square, and thus a vanishing degree. Then it must be trivial. □
6.2. Any oriented 3-manifold is parallelizable
In this subsection, we prove the following standard theorem. The spirit of our proof is the same as the Kirby proof in [53. P. 46]. But instead of assuming familiarity with the obstruction theory described by Steenrod in [54. Part III], we use this proof as an introduction to this theory.
Theorem 9. [Stiefel]. Any oriented 3-manifold is parallelizable.
Lemma 30. The restriction of the tangent bundle TM to an oriented 3-manifold M to any closed (non-necessarily orientable) surface S immersed in M is trivializable.
Proof. Let us first prove that this bundle is independent of the immersion. It is the direct sum of the tangent bundle to the surface and of its normal one-dimensional bundle. This normal bundle is trivial when S is orientable, and its unit bundle is the 2-fold orientation cover of the surface, otherwise. (The orientation cover of S is its 2-fold orientable cover, which is trivial over annuli embedded in the surface). Then since any surface S can be immersed in R3, the restriction TM|S is the pull-back of the trivial bundle of R3 by such an immersion, and it is trivial. □
Then using Stiefel — Whitney classes, the proof of Theorem 9 quickly goes as follows. Let M be an orientable smooth 3-manifold, equipped with a smooth triangulation. (A theorem of Whitehead proved in the Munkres book [55] ensures the existence of such a triangulation.) By definition, the first Stiefel—Whitney class k^MM) hH\M\Z/2Z = n0(GL(M3))) seen as a map from n4(M) to Z / 2Z maps the class of a loop c embedded in M to 0 if TM|c is orientable and to 1 otherwise. It is the obstruction to the existence of a trivialization of TM over the one-skeleton of M. Since M is orientable, the first Stiefel — Whitney class w,(TM) vanishes and TM can be trivialized over the one-skeleton of M. The second Stiefel—Whitney class w2(TM) e H2(M;Z /2Z = rc^GL^K3))) seen as a map from H2(M;Z/21) to Z/ 2Z maps the class of a connected closed surface S to 0 if TM|S is trivializable and to 1 otherwise. The second Stiefel — Whitney class w2(TM) is the obstruction to the existence of a trivialization of TM over the two-skeleton of M, when w1(TM) = 0. According to the above lemma, w2(TM) = 0, and TM can be trivialized over the two-skeleton of M. Then since n2(GL+ (M3)) = 0, any par-allelization over the two-skeleton of M can be extended as a parallelization of M. □
We detail the involved arguments below without mentioning Stiefel — Whitney classes, (actually by almost defining w2(TM)). The elementary proof below can be thought of as an introduction to the obstruction theory used above.
Elementary proof of Theorem 9. Let M be an oriented 3-manifold. Choose a triangulation of M. For any cell c of the triangulation, define an arbitrary trivialization tc : c x M3 ^ TM|c such
that t induces the orientation of M. This defines a trivialization t(0) : M(0) x M3 ^ TM of
c |M(0)
M over the 0-skeleton M(0) of M. Let Ck(M) be the set of k-cells of the triangulation. Every cell is equipped with an arbitrary orientation. For an edge e e C1(M) of the triangulation, on de, t(0) reads t(0) = Te o ym(ge) for a map ge : de ^ GL+ (M3). Since GL+ (M3) is connected, ge extends to e, and t(1) = Te o yM(ge) extends t(0) to e. Doing so for all the edges extends t(0) to a trivialization t of the one- skeleton M(1) of M.
For an oriented triangle t of the triangulation, on dt, t(1) reads t(1) = Tt o yM (gt) for a map gt : dt ^ GL+ (M3). Let E(t,t(1)) be the homotopy class of gt in (n/GL+OR3)) = n4(SO(3)) = Z/2Z, E(t,t(1)) is independent of Tt. Then E(., t(1)) : C2(M) ^Z/2Z is a cochain. When E(.,t(1)) = 0, t(1) may be extended to a trivialization t(2) over the two-skeleton of M, as before.
Since n2(GL+ (M3)) = 0, t(2) can next be extended over the three-skeleton of M, that is over M.
Let us now study the obstruction cochain E(., t(1)) whose vanishing guarantees the existence of a parallelization of M.
If the map ge associated to e is changed to d(e)ge for some d(e): (e,de) ^(GL+(R3),1) for every edge e, define the associated trivialization t(1)', and the cochain D(t(1), t(1)') ^Z / 2Z
that maps e to the homotopy class of d(e). Then (E(.,T(i)')-E(.,T(i)) ) is the coboundary of D(t(1), t(1)').
Let us show that E(,t(1)) is a cocycle. Consider a 3-simplex T, then t(0) extends to T. Without loss of generality, assume that tt coincides with this extension, that for any face t of T, Tt is the restriction of tt to t, and that the above t(1)' coincides with tt on the edges of dT. Then E( ,T(i)')(dT) = 0. Since a coboundary also maps dT to 0, E(,T(i)')(dT) = 0.
Now, it suffices to prove that the cohomology class of E(,t(1)) (which is actually w2(TM)) vanishes in order to prove that there is an extension T(i)' of t(0) on M(i) that extends on M.
Since H2(M;Z/2Z) = Hom(H2(M;Z/2Z));Z/2Z), it suffices to prove that E( ,t(1)) maps any 2-dimensional Z/ 2Z-cycle C to 0.
We represent the class of such a cycle C by a non-necessarily orientable closed surface S as follows. Let N(M(0)) and N(M(i)) be small regular neighborhoods of M(0) and M(i) in M, respectively, such that N(M(1)) n (M \ N(M(0)))is a disjoint union, running over the edges e, of solid cylinders Be identified with ]0,i[xD2. The core ]0,i[x{0} of Be =]0,i[xD2 is a connected part of the interior of the edge e. (N(M(i)) is thinner than N(M(0)).)
Construct S in the complement of N(M(0)) u N(M(1)) as the intersection of the support of C with this complement. Then the closure of S meets the part [0,i] x S1 of every Be as an even number of parallel intervals from {0} x S1 to {i} x S1. Complete S in M \ N(M(0)) by connecting the intervals pairwise in Be by disjoint bands. After this operation, the boundary of the closure of S is a disjoint union of circles in the boundary of N(M(0)), where N(M(0)) is a disjoint union of balls around the vertices. Glue disjoint disks of N(M(0)) along these circles to finish the construction of S.
Extend t(0) to N(M(0)), assume that T(i) coincides with this extension over M(1) n N(M(0)), and extend t to N(M(i)). Then TMS is trivial, and we may choose a trivialization ts of TM over S that coincides with our extension of t(0) over N(M(0)), over S n N(M(0)). We have a cell decomposition of (S, S n N(M(0))) with only i-cells and 2-cells, where the 2-cells of S are in one-to-one canonical correspondence with the 2-cells of C, and one-cells bijectively correspond to bands connecting two-cells in the cylinders Be. These one-cells are equipped with the trivialization of TM induced by t(1). Then we can define 2-dimensional cochains Es(.,t(1)) and Es(.,ts) from C2(S) to Z/2ft as before, with respect to this cellular decomposition of S, where (Es(-,t(1)) — Es(.,ts)) is again a coboundary and Es(,ts) = 0 so that ES(C,t(1)) = 0, and since E(C, t(1)) = ES(C, t(1)), E(C, t(1)) = 0 and we are done. □
6.3. The homomorphism induced by the degree on [(M, 3M),(SO(3),i)]
Let S be a non-necessarily orientable closed surface embedded in the interior of M, and let t be a parallelization of M. We define a twist g(S, t) e C0 ((M, dM),(SO(3),i)) below.
The surface S has a tubular neighborhood N(S), which is a [-1,1]-bundle over S that admits (orientation-preserving) bundle charts with domains [-i,i] x D for disks D of S so that the changes of coordinates restrict to the fibers as ± Identity. Then
g(S,t) : (M,dM) ^ (GL+ (M3),i)
is the continuous map that maps M\N(S) to i such that g(S, T)((t, s) e [-i,i] x D) is the rotation with angle n(t+i) and with axis p2(T-i(vs) = (s,p2(T-i(vs)))) where vs = T(0s)([-i,i] x s) is the tangent vector to the fiber [-i,i] x s at (0,s). Since this rotation coincides with the rotation with opposite axis and with opposite angle n(1-t), our map g(S, t) is a well-defined continuous map.
Clearly, the homotopy class of g(S, t) only depends on the homotopy class of t and on the isoto-
i
py class of S. When M=B3, when t is the standard parallelization of M3, and when — S2 denotes the sphere i 6B3 inside B3, the homotopy class of g ^^ s2, coincides with the homotopy class of p.
Lemma 31. Hl(g(S, t);Z /2Z) is the mod 2 intersection with S. The map H4(.;Z/2Z) from [(M,dM),(SO(3),1)] to Hl(M,dM;Z/2Z) is onto. Proof. The first assertion is obvious, and the second one follows since H4(M,dM;Z/2Z) is the Poincare dual of H2(M; Z / 2Z) and since any element of H2(M; Z / 2Z) is the class of a closed surface. □
Lemma 32. The degree is a group homomorphism deg: [(M,dM),(SO(3),1)] ^ Z and deg (Pm (B3)k) = 2k.
Proof. It is easy to see that deg(fg) = deg(f) + deg(g) when f or g is a power of [pM(B3)]. Let us prove that deg(f2) = 2deg(f) for any f. According to Lemma 31, there is an unoriented embedded surface Sf of the interior of C such that Hi(f; Z/21) = Hx(g(Sf,T);2,/2Z) for some trivialization t of TM. Then, according to Lemma 29, fg(Sf, t)-1 is homotopic to some power of PM(B3), and we are left with the proof that the degree of g2 is 2deg(g) for g = g(Sf, t). This can easily be done by noticing that g2 is homotopic to g(S(2), t) where S(2) is the boundary of the tubular
111
neighborhood of Sf. In general, deg(fg) = ^deg((fg)2) = ^deg(f2ff2) = ^(g(f2) +deg(g2)),
and the lemma is proved. □
Lemmas 29 and 32 imply the following lemma.
Lemma 33. The degree induces an isomorphism deg: [(M, dM),(SO(3),1)] ®Z Q ^ Q. Any
group homomorphism ^ : [(M,dM),(SO(3),1)]
1
reads ^ ^(pM (B3))deg.
6.4. First homotopy groups of the groups SU(n)
Let K = M or C. Let n e N. The stabilization maps induced by the inclusions
i: GL(Kn) ^ GL(K © Kn)
g ^ (i(g): (x, y) ^ (x, g(y)))
will be denoted by i. Elements of GL(kn) are represented by matrices whose columns contain the coordinates of the images of the basis elements, with respect to the standard basis of Kn.
See S3 as the unit sphere of C2 so that its elements are the pairs (zx,z2) of complex numbers such that | z1 |2 + | z2 |2= 1. The group SU(2) is identified with S3 by the homeomorphism
m.
c .
S3
SU(2)
(zu z2) ^
Z2 Z1
so that the first non trivial homotopy group of SU(2) is n3(SU(2)) = Z[mC ]. The long exact sequence associated to the fibration
2n-1
SU(n - 1)^SU(n) ^ S
shows that i*n : nj(SU(2)) ^ nj(SU(n + 2)) is an isomorphism for j < 3 and n > 0, and in particular, that n..(SU(4)) = {1} for j < 2 and n3(SU(4)) = Z[i2(mrC)] where i2(mrC) is the following map
i2(mC): (S3 c C2)
(z4, z2)
SU(4) 1 0 0 0 1 0 0 0 z 0 0 z,
0 0
-Z2 Z1
6.5. Definition of relative Pontrjagin numbers
Let M0 and M1 be two compact connected oriented 3-manifolds whose boundaries have collars that are identified by a diffeomorphism. Let t0 : M0 x R3 ^ TM0 and t1 : M1 x R3 ^ TM1 be two parallelizations (which respect the orientations) that agree on the collar neighborhoods of dM0 = dM1. Then the relative Pontrjagin number p1(t0, t1) is the Pontrjagin obstruction to extending the trivialization of TW ® C induced by t0 and t1 across the interior of a signature 0 cobordism W from M0 to M1. Details follow.
Let M be a compact connected oriented 3-manifold. A special complex trivialization of TM is a trivialization of TM ® C that is obtained from a trivialization tm : M x R3 ^ TM that induces the orientation of M by composing (tM = Tm ®R C): M x C3 ^ TM ® C
y(G): M x C3 ^ M x C3
(x, y) ^ (x,G(x)(y)) for a map G : M ^ SL(3, C). The definition and properties of relative Pontrjagin numbers, which are given with more details below, are valid for pairs of special complex trivializations.
The signature of a 4-manifold is the signature of the intersection form on its H2( ;R) (number of positive entries minus number of negative entries in a diagonalised version of this form). Also recall that any closed oriented three-manifold bounds a compact oriented 4-dimensional manifold whose signature may be arbitrarily changed by connected sums with copies of cP2 or -CP2. A cobordism from M0 to M1 is a compact oriented 4-dimensional manifold W with corners whose boundary dW is equal to -M0 ^m^oxSM0 (-[0,1] x5M0) u^^m,, Mp and is identified with an open subspace of one of the products [0,1[xM0 or ]0,1] x M1 near dW , as Fig. 3 suggests.
{0} x M0 = M0
[0,1] x (-dMoY
Fig. 3
{1} x Mi = Mi
Let W = W4 be such a cobordism from M0 to M1, with signature 0. Consider the complex 4-bundle TW ® C over W. Let V be the tangent vector to [0,1] x {pt} over dW (under the identifications above), and let t(t0,t1) denote the trivialization of TW ® C over dW that is obtained by stabilizing either t0 or t1 into v © t0 or v © t1 . Then the obstruction to extending this trivialization to W is the relative first Pontrjagin class p1(W; t(t0, t1))[W, dW] of the trivialization, which belongs to H4(W, dW; Z = n3(SU(4))) = Z[W, dW].
Now, we specify our sign conventions for this Pontrjagin class. They are the same as in [56]. In particular, p1 is the opposite of the second Chern class c2 of the complexified tangent bundle. See [56. P. 174]. More precisely, equip M0 and M1 with Riemannian metrics that coincide near dM0, and equip W with a Riemannian metric that coincides with the orthogonal product metric of one of the products [0,1] x M0 or [0,1] x M1 near dW. Equip TW ® C with the associated hermitian structure. The determinant bundle of TW is trivial because W is oriented, and det(TW ® C) is also trivial. Our parallelization t(t0, t1) over dW is special with respect to the trivialization of det(TW ® C) . Up to homotopy, assume that t(t0, t1) is unitary with respect to the hermitian structure of TW ® C and the standard hermitian form of C4. Since ni(SU(4)) = {0} when i <3, the trivialization t(t0,t1) extends to a special unitary trivialization t outside the interior of a 4-ball B4 and defines
t : S3 x C4 ^ (TW ® C) 3
over the boundary S3 = dBi of this 4-ball B4. Over this 4-ball B4, the bundle TW ® C admits a trivialization tb : B4 x C4 ^ (TW ® C)^. Then xB4 ° t(v e S3, w e C4) = (v, |(v)(w)), for a map
| : S3 ^ SU(4) whose homotopy class reads
[|] = -p4(W;t(t0,x1))[i2(mrC)] e n,(SU(4)).
Define p4(x0, T ) = p4(W; t(v t,)).
Proposition 9. The first Pontrjagin number p4(x0, t4) is well-defined by the above conditions.
Proof. According to the Nokivov additivity theorem, if a closed (compact, without boundary) 4-manifold Y reads Y = Y + uX Y~ where Y + and Y- are two 4-manifolds with boundary, embedded in Y that intersect along a closed 3-manifold X (their common boundary, up to orientation) then
signature(Y) = signature( Y+) + signature( Y").
According to a Rohlin theorem (see [57] or [58. P. 18]), when Y is a compact oriented 4-manifold without boundary, p4(Y) = 3 signature( Y).
We only need to prove that p4(x0, t4) is independent of the signature 0 cobordism W. Let WE be a 4-manifold of signature 0 bounded by (-dW). Then W WE is a 4-dimensional manifold without boundary whose signature is (signature( WE) + signature(W) = 0) by the Novikov additivity theorem. According to the Rohlin theorem, the first Pontrjagin class of W WE is also zero. On the other hand, this first Pontrjagin class is the sum of the relative first Pontrjagin classes of W and WE with respect to t(t0,t4). These two relative Pontrjagin classes are opposite and therefore the relative first Pontrjagin class of W with respect to t(t0, Tj) does not depend on W.
Similarly, it is easy to prove the following proposition.
Proposition 10. Under the above assumptions except for the assumption on the signature of the cobordism W, p4(t0, t4) = p1(W; t(t0, t4)) - 3 signature( WE).
6.6. On the groups SO(3) and SO(4)
In this subsection, we describe n3(SO(4)) and the natural maps from n3(SO(3)) to n3(SO(4)) and to n3(SU(4)).
The quaternion field H is the vector space_C © cj equipped with the multiplication that maps (z4+ z2j, zj+z2j) to (z1z'1 - z2z'2) + (z2z\ + zlz'2)j, and with the conjugation that
maps (z1 + z2 j) to z1 + z2 j = z1 - z2 j . The norm of (z1 + z2 j) is the square root of
| z4 |2 + | z2 |2= (z4 + z2 j)z1 + z2 j, it is multiplicative. Setting k = ij, (1, i, j, k) is an orthogonal basis of H with respect to the scalar product associated to the norm. The unit sphere of H is the sphere S3, which is equipped with the corresponding group structure. There are two group morphisms from S3 to SO(4) induced by the multiplication in H,
mt : S3 ^ (SO(H) = SO(4))
x ^ mf (x) : v ^ x.v
m: S3 ^ SO(H)
y ^ (mr(y) : v ^ v.y).
Together, they induce the group morphism
S3 x S3 ^ SO(4) (x, y) ^ (v ^ x.v.y). The kernel of this group morphism is Z/2Z(-1,-1) so that this morphism is a two-fold covering. In particular, n3(SO(4)) = Z[mt] © Z[m~r].
For K = R or C and n e N, the K (euclidean or hermitian) oriented vector space with the direct orthonormal basis (v4,...,vn) is denoted by K <v1,...,vn>. There is also the following group morphism
p : S3 ^ SO(R < i, j, k >) = SO(3) X ^ (v ^ (v ^ x.v.x))
whose kernel is Z/ 2Z(-1). This morphism p is also a two-fold covering.
Lemma 34. This definition of p coincides with the previous one,up to homotopy.
Proof. It is clear that the two maps coincide up to homotopy, up to orientation since both classes generate n3(SO(3)) = Z. We take care of the orientation using the outward normal first convention to orient boundaries, as usual. An element of S3 reads cos(0) + sin(0)v for a unique 0 e [0, n] and a unit quaternion v with real part zero, which is unique when 0 £ {0, n}. In particular, this defines a diffeomorphism ^ from ]0, n[xS2 to S3\{-1,1}. We compute the degree of ^ at ^(n /2,i). The space H is oriented as R © R(i, j, k>, where R(i, j, k> is oriented by the outward normal to S2, which coincides with the outward normal to S3 in R4, followed by the orientation of S2. In particular since cos is an orientation-reversing diffeomorphism at n / 2, the degree of ^ is 1 and ^ preserves the orientation. Now (cos(0) + sin(0)v)w(cos(0) + sin(0)v) = R(0,v)(w) where R(0,v) is a rotation with axis v for any v. Since R(0,i)(j) = cos(20)j + sin(20)k, the two maps p are homotopic. One can check that they are actually conjugate. □
Define
mr : S3 ^ (SO(H) = SO(4)) y ^ (mr (y) : v ^ v.y).
Lemma 35. In n3(SO(4)) = Z[mf ] © Z[mr], i([p]) = [mf ] + [mr] = [mt] - [mr].
Proof. The n3-product in n3(SO(4)) coincides with the product induced by the group structure of SO(4). □
Lemma 36. Recall that mr denotes the map from the unit sphere S3 of H to SO(H) induced by the right-multiplication. Denote the inclusions SO(n) c SU(n) by c. Then in n3(SU(4)), c* ([mr]) = 2[i2(mrC )].
Proof. Let H + IH denote the complexification of R4 = H = R(1,i, j, k>. Here, C = R © IR. When x e H and v e S3, c(mr)(v)(lx) = Ix.v, and I2 = -1. Let s = ±1, define
C2(s) = C ^ ^ (1 + SIi),f( j + elkjj.
Consider the quotient C4/C2(s). In this quotient, Ii = -s1, Ik = -sj, and since I2 = -1, I1 = si and Ij = sk . Therefore this quotient is isomorphic to H as a real vector space with its complex structure I = si. Then it is easy to see that c(mr) maps C2(s) to 0 in this quotient. Thus c(mr)(C2(s)) = C2(s). Now, observe that H + IH is the orthogonal sum of C2(-1) and C2(1). In particular, C2(s) is isomorphic to the quotient C4/C2(-s), which is isomorphic to (H; I = -si) and c(m ) acts on it by the right multiplication. Therefore, with respect to the
basis f (1 - Ii,j - Ik,, + Ii,j + Ik), c(m, Xz_ + , j) reads
z -Z2 0 0
z2 Z1 0 0
0 0 Z1 = x1 - Iy{ -z2
0 0 Z2 z1= x1 + Iy1
Therefore, the homotopy class of c(mr) is the sum of the homotopy classes of
m; (z1, z2) 0
(z1 + z2 j) ^
and
(zi + z2 j) ^
1 0
0 m; o t(z., z2)
where i(z1, z2) - (z1, z2). Since the first map is conjugate by a i2(mrc )
it is homotopic to ), and since i induces the identity on n3(S3), the second map is homo-
topic to i2(m;), too. □
The following lemma finishes to determine the maps c* : n3(SO(4)) ^ n3(SU(4)) and c*i* : n3(SO(3)) ^_3№(4)). Lemma 37. c*([mr]) - c*([mt]) - -2[i2(mrc)], c*(i*([,5])) - -4[i2(mrc)].
Proof. According to Lemma 35, i*([p]) - [mt] + [mr] - [mt] - [mr]. Using the conjugacy of
quaternions, mf (v)(x) - v.x - x.v - mr(v)(x). Therefore mt is conjugated to mr via the conjugacy of quaternions, which lies in (O(4) c U(4)).
Since U(4) is connected, the conjugacy by an element of U(4) induces the identity on *3(SU(4)). Thus, c*(K])- c*([mr])- -c*([mr]), and c*(i*([p])) - -2c*([mr]).
6.7. Relating the relative Pontrjagin number to the degree
We finish proving Theorem 2 by proving the following proposition.
Proposition 11. Let M0 and M be two compact connected oriented 3-manifolds whose boundaries have collars that are identified by a diffeomorphism. Let t0 : M0 x C3 ^ TM0 ® C and t : M x C3 ^ TM ® C be two special complex trivializations (which respect the orientations) that coincide on the collar neighborhoods of dM0 = dM. Let [(M,dM),(SU(3),1)] denote the group of homotopy classes of maps from M to SU(3) that map dM to 1. For any g : (M, dM) ^ (SU(3),1), define
y(g): M x C3 ^ M x C3
(x, y) ^ (x, g(x)(y))
then p1(x0,t o y(g)) - p1(x0,t) = p4(x,t o y(g)) = -p4(x o y(g),t) = p\(g) is independent from t0 and t, p[ induces an isomorphism from the group [(M, dM),(SU(3),1)] to Z, and, if g is valued in SO(3), then p[(g) = 2deg(g).
In order to prove this proposition, we first prove the following lemma.
Lemma 38. Under the hypotheses of Proposition 11, (p4(t0, t o y(g)) - p4(t0, t)) is independent from t0 and t.
Proof. Indeed, (p4(t0, t o y(g)) - p4(t0, t)) can be defined as the obstruction to extending the following trivialization of the complexified tangent bundle to [0,1] x M restricted to the boundary. This trivialization is T[0,1] ©t on ({0} x M) u([0,1] x dM) and T[0,1] ©t o y(g) on {1} x M. But this obstruction is the obstruction to extending the map g from d([0,1] x M) to SU(4) that maps ({0} x M) u ([0,1] x dM) to 1 and that coincides with i(g) on {1} x M, regarded as a map from d([0,1] x M) to SU(4), over ([0,1] x M). This obstruction, which lies in n3(SU(4)) since ni(SU(4)) = 0, for i <3, is independent of t0 and t.
Proof of Proposition 11. Lemma 38 guarantees that p[ defines two group homomorphisms to Z from [(M,dM),(SU(3),1)] and from [(M,dM),(S0(3),1)]. Since n/(SU(3)) is trivial for i <3 and since n3(SU(3)) = Z, the group of homotopy classes [(M,dM),(SU(3),1)] is generated by the class of a map that maps the complement of a 3-ball B to 1 and that factors
through a map that generates n3(SU(3)). By definition of the Pontrjagin classes, p sends such a generator to ±1 and it induces an isomorphism from [(M,3M),(SU(3),1)] to Z.
According to Lemma 29 and to Lemma 23, the restriction of p to [(M,3M),(SO(3),1)]
must read p\(pM(B3))^^, and we are left with the proof of the following lemma.
Lemma 39. p (pM (B3)) = 4.
Let g = pM(B3), we can extend g (defined in the proof of Lemma 38) by the constant map with value 1 outside [s,1] x B3 = B4 and, in n3(SU(4)) [c(g^gi)] = -p4(x,t ° y(g))[i2(mrc)]. Since g ^ is homotopic to c ° i(p), Lemma 37 allows us to conclude. □
7. Other complements
7.1. More on low-dimensional manifolds
Piecewise linear (or PL) n-manifolds can be defined as the Ci-manifolds of Subsection 1.2 by replacing Cl with piecewise linear (or PL).
When n < 3, the above notion of PL-manifold coincides with the notions of smooth and topological manifold, according to the following theorem. This is not true anymore when n >3. See [59].
Theorem 10. When n < 3, the category of topological n-manifolds is isomorphic to the category of PL n-manifolds and to the category of Cr n-manifolds, for r = 1,..., ro.
For example, according to this statement, which contains several theorems (see [31]), any topological 3-manifold has a unique C" -structure. Below n = 3.
The equivalence between the categories Ci, i = 1,2,., ro follows from work of Whitney in 1936 [60]. In 1934, Cairns [61] provided a map from the C4-category to the PL category, which shows the existence of a triangulation for C'-manifolds, and he proved that this map is onto [62. Theorem III] in 1940. Moise [63] proved the equivalence between the topological category and the PL category in 1952. This diagram was completed by Munkres [64. Theorem 6.3] and Whitehead [65] in 1960 by their independent proofs of the injectivity of the natural map from the C4-category to the topological category.
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About the author
Christine Lescop, research director, Institut Fourier, French National Centre for Scientific Research, University Joseph Fourier, Saint-Martin-d'Heres, France. Christine. Lescop@ ujf-grenoble.fr.
Bulletin of Chelyabinsk State University. 2015. № 3 (358).
Mathematics. Mechanics. Informatics. Issue 17. Р. 67-117.
ВВЕДЕНИЕ В ТЕОРИЮ ИНВАРИАНТОВ КОНЕЧНОГО ТИПА УЗЛОВ И ТРЕХМЕРНЫХ МНОГООБРАЗИЙ, ОПРЕДЕЛЯЕМЫХ КАК ЧИСЛО КОНФИГУРАЦИЙ В ГРАФЕ
К. Лескоп
Концепция инвариантов конечного типа для узлов была предложена в 90-х гг. в работах Васильева, Гусарова и Бар-Натана с целью классификации инвариантов узлов вскоре после появления многочисленных квантовых инвариантов узлов. Эта очень полезная концепция была расширена Отсуки до случая инвариантов трехмерных многообразий.
В статье показывается, как определить инварианты конечного типа для узлов и трехмерных многообразий путем подсчета конфигураций графа в трехмерных многообразиях. Мы следуем идеям Виттена и Концевича.
Число зацеплений является простейшим инвариантом конечного типа для двухкомпонентных зацеплений. Он определяется несколькими эквивалентными способами в первом разделе. В качестве важного примера приводится его определение как алгебраическое пересечение тора и 4-цепи, называемое пропагатором в конфигурационном пространстве.
Во втором разделе мы вводим простейший инвариант конечного типа для трехмерных многообразий — инвариант Кассона (или ©-инвариант) целочисленных гомологических 3-сфер. Он определяется как алгебраическое пересечение трех пропагаторов в одном и том же двухточечном конфигурационном пространстве.
В третьем разделе описано общее понятие инварианта конечного типа и введены соответствующие пространства диаграмм Фейнмана — Якоби.
В разделах 4 и 5 мы даем набросок оригинальной конструкции, основанной на интегралах конфигурационного пространства универсальных инвариантов конечного типа для зацеплений в рациональных гомологических сферах, а также формулируем несколько нерешенных проблем. Наша конструкция обобщает известные конструкции для зацеплений в М3 и для рациональных гомологических 3-сфер, что делает ее более гибкой.
В разделе 6 детально описываны необходимые свойства параллелизаций трехмерных многообразий и соответствующих классов Понтрягина.
Ключевые слова: узлы, трехмерные многообразия, инварианты конечных типа, гомологические 3-сферы, число зацеплений, тета-инвариант, инвариант Кассона-Уолкера, диаграммы Фейнмана-Якоби, расширение теории Черна-Саймонса, интегралы конфигурационного пространства, параллелизация трехмерных многообразий, первый класс Понтрягина.
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Сведения об авторе
Лескоп Кристина, руководитель исследованиями, Институт Фурье, Национальный центр научных исследований Франции, Университет Жозефа Фурье, Сен-Мартен-д'Эр, Франция. Christine.Lescop@ujf-grenoble.fr.
