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1Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 3, pp. 371-381. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd230905
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 37C15, 37C27, 37D15
Topology of Ambient 3-Manifolds of Non-Singular Flows with Twisted Saddle Orbit
O. V. Pochinka, D. D. Shubin
In the present paper, nonsingular Morse -Smale flows on closed orientable 3-manifolds are considered under the assumption that among the periodic orbits of the flow there is only one saddle and that it is twisted. An exhaustive description of the topology of such manifolds is obtained. Namely, it is established that any manifold admitting such flows is either a lens space or a connected sum of a lens space with a projective space, or Seifert manifolds with a base homeomorphic to a sphere and three singular fibers. Since the latter are prime manifolds, the result obtained refutes the claim that, among prime manifolds, the flows considered admit only lens spaces.
Keywords: nonsingular flows, Morse-Smale flows, Seifert fiber space
1. Introduction and formulation of results
In the present paper, we consider NMS-flows fl, that is, nonsingular (without fixed points) Morse-Smale flows defined on closed orientable 3-manifolds M3. The nonwandering set of such flows consists of a finite number of periodic hyperbolic orbits. It is known from Asimov's work [1] that the ambient manifold in this case has a round handle decomposition. However, in the case of a small number of periodic orbits, the topology of the manifold can be significantly refined. For example, only lens spaces are ambient for NMS-flows with exactly two periodic orbits. Moreover, in [2] it is proved that for every lens space there are exactly two equivalence classes of such flows, except for the 3-sphere S3 and the projective space RP3, on which there is one equivalence class.
Received December 26, 2022 Accepted August 25, 2023
This work was performed at the Saint Petersburg Leonhard Euler International Mathematical Institute and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2022-287).
Olga V. Pochinka olga-pochinka@yandex.ru Danila D. Shubin schub.danil@yandex.ru
National Research University "Higher School of Economics" ul. Bolshaya Pecherskaya 25/12, Nizhny Novgorod, 603155 Russia
In [3], it is stated that the lens space is also the only prime (homeomorphic to S2 x S1 or irreducible — any cylindrically embedded 2-sphere bounds the 3-ball) 3-manifold which is ambient for NMS-flows with a unique saddle periodic orbit. However, this is incorrect. In the previous work of one of the authors [4], NMS-flows with exactly three periodic orbits (attractive, repelling and saddle) are constructed on a countable set of pairwise nonhomeomorphic mapping tori that are not lens spaces. Moreover, in [5] necessary and sufficient conditions for the topological equivalence of such flows are obtained.
In this paper, we recognize the topology of all orientable 3-manifolds that admit NMS-flows with exactly one saddle periodic orbit, assuming that it is twisted (its invariant manifolds are nonorientable).
Let us proceed to the formulation of the results.
Let M3 be a connected closed orientable 3-manifold, f1: M3 — M3 an NMS-flow and O — its periodic orbit. In the neighborhood of the hyperbolic periodic orbit O, the flow can be simply described (up to topological equivalence). Namely, there exist a linear diffeomorphism of the plane, given by the matrix with positive determinant and real eigenvalues with absolute value different from one, and a tubular neighborhood VQ homeomorphic to the solid torus D2 x S1, in which the flow is topologically equivalent to the suspension over this diffeomorphism (see, for example, [6]). If both eigenvalues are greater (less) than one in absolute value, then the corresponding periodic orbit is called repelling (attractive) and saddle otherwise. In this case, a saddle orbit is called twisted if both eigenvalues are negative and untwisted otherwise.
Let T0 = dV0. Let us choose meridian M0 c T0 (a null-homotopic curve on V0 and essential on TQ) and longitude L0 c TQ (the curve homologous in the VG to the orbit O). We assume that the meridian M0 is oriented so that the pair of oriented curves M0, L0 determines the outer side of the solid torus boundary. Thus, the homotopy types (L0) = (1, 0), (M0) = = (0, 1) of knots L0, M0 are generators of the homotopy types (K) of oriented knots K on torus TO, that is,
(K) = (lo ,mo) = lo (Lo ) + mo (M0), (1.1)
where l0, m0 G Z are numbers of twists of the oriented knot K around the parallel and the meridian, respectively. Note that the choice of the longitude does not influence the following reasoning, since only the remainder of the division of l by m matters in the subsequent discussion.
Consider the class G- (M3) of NMS-flows f1: M3 — M3 with a unique saddle orbit, assuming that it is twisted. Since the ambient manifold M3 is the union of the stable (unstable) manifolds of all its periodic orbits, the flow f G G- (M3) must have at least one attracting and at least one repelling orbit. In Section 3, we will prove the following fact.
Lemma 1. The nonwandering set of any flow f1 G G- (M3) consists of exactly three periodic orbits S, A, R, saddle, attracting and repelling, respectively.
Since the flow f in the neighborhood of a periodic orbit is equivalent to a suspension over the linear diffeomorphism, the stable and unstable manifolds of these orbits have the following topology:
• = W $ = R x S1 (open Moebius strip);
• WA ^ WR ^ R2 x S1;
• WA = WR = S1.
This fact and Lemma 1 immediately imply the following proposition (for more details,
see [5]).
Proposition 1. The ambient manifold M3 of any flow f1 £ G- (M3) is represented as the union of three solid tori:
M3 = VA U Vs U VR
with disjoint interiors being tubular neighborhoods of A, S, R orbits, respectively, with the following properties:
Ts = dVs is the union of tubular neighborhoods Tg, TS of knots Kg = Wg H Ts, Kss = W's n Ts, respectively, such that Tg n Tl = dTg H dT;!;
the torus Ta = dVa is the union of the annulus Tg and a compact surface T (an annulus or disjoint union of a handle with a disk) with disjoint interiors, and the knot Kg has homotopy type
(Kg) = (¡a, mA) with generators La, Ma, respectively;
the torus Tr = dVR is the union of the annulus TS and the surface T with disjoint interiors, and the knot KSs has homotopy type
(KS) = (¡R, mR) with generators LR, Mr, respectively.
(a) essential
(b) inessential
Fig. 1. Knot K
Thus, both knots Kg c Ta, KS C TR are either inessential or essential (see Fig. 1). For every flow f1 £ G- (M3) we determine a quadruple of integers
Cft = (l1, mx, l2, m2)
as follows:
• if the knots Kg, Kss are essential on tori TA, TR, then
Cft = (lr, mR, 1a, mA);
• if the knots Kg, KsS are inessential on tori Ta, Tr, then
Cft = (0, 2, I2, m2),
where (l2, m2) is the homotopy type of the knot on torus TR which is the meridian on torus Ta.
Note that the class G- (M3) is not empty, because by [5] every quadruple C = (l1, m1 ,l2, m2) with gcd(li, mi) = 1, i = 1,2 and quadruple C = (0, 2, l2, m2) with gcd(l2, m2) = 1 are realizable by a flow ff G G3 (M3) such that C = Cft.
The main result of the paper is the following theorem (all the necessary information about the objects mentioned below is given in Section 2).
Theorem 1. Ambient manifolds of the flows in G- (M3) are lens spaces Lpq, connected sums of the form Lp, #RP3 and Seifert manifolds of the form M (S2, (2, 1), (a1, fi^, (a2, f2)). Namely, let the flow f1 G G- (M3) correspond to the collection Cf t = (l1, m1, l2, m2). Then
1) if l1 = 0 and l2 = 0, then M3 is homeomorphic to the manifold Ll #RP3;
2) if l1 = 0 and l2 = 0, then M3 is homeomorphic to the manifold Li m^#RP3;
3) if l1 = 0 and l2 = 0, then M3 is homeomorphic to S2 x S1#RP3;
4) if 1^1 = 1 and \l2\ > 1, then M3 is homeomorphic to the lens space Lpq, where p = 2f2 — l2b, q = ^til _ p2t f32m,2 = i (mod l2),b = 1 (mod 2);
5) if \l2\ = 1 and \l1 \ > 1, then M3 is homeomorphic to the lens space Lp q, where p = 2f1 — l1b, q = ^til _ pit piTni = i (mod l{), b = 1 (mod 2);
6) if \l1l2\ = 1, then M3 is homeomorphic to the lens space Lb 2, b = 1 (mod 2);
7) if |l1\ > 1 and \l2\ > 1, then M3 is homeomorphic to the prime Seifert manifold M (S2, (2, 1), (l1, f1), (l2, f2)), fimi = 1 (mod li), i = 1, 2 and is not homeomorphic to any lens space.
2. Necessary information on the topology of 3-manifolds
2.1. Lens spaces
Everywhere below we assume that generators of homotopy types of knots on boundary dV of the standard solid torus V = D2 x S1 are meridian M = (dB2) x {y}, y G S1 with homotopy type (0, 1) and parallel L = {x} x S1, x G dB2 with homotopy type (1, 0).
Lens space is a three-dimensional manifold Lp q = V1 U V2, which is the result of gluing
p'q j
together two copies of the solid torus V1 = V, V2 = V by some homeomorphism j: dV1 — dV2 such that j*((0, 1)) = (p, q).
Proposition 2 ([7]). Two lens spaces Lp, q, Lp, , are homeomorphic (up to preserving the numbering of copies) if and only if p = ±p', q = ±q' (mod \p\).
2.2. Dehn surgery along knots and links
Suppose the following data are given:
1) a closed 3-manifold M;
2) a knot y C M;
3) a tubular neighborhood UY of y with standard generators on dUY: meridian MY and longitude Ly ;
4) a homeomorphism h: dV — dUY inducing an isomorphism such that h*((0, 1)) = (ft, a). A manifold
MYhh = (M \ int UY) U V
is called the manifold obtained from M by Dehn surgery along the knot y.
Naturally, the manifold M is restored from MY/h by inverse surgery. Namely, we denote by PYhh: (M \ int Uy) U V — Mlh the natural projection. Let y = PY,h ({0} x S1), U~ = PYh(V), h = PY,hh-1: dUY dU~. Then
M = (M^-. (2.1)
The following assertions follow directly from the relation (2.1).
Proposition 3. Let y C M be equipped with ¡3, a. Then
M = (my
where Y is equipped with —3, £ satisfying + vfl = 1.
Proposition 4. Let Lp,q = V1 U V2, where j*((0, 1)) = (p, q) and S3 = V1 U V2, where jo*((0, 1)) = (1, 0). Then
Lp,q — SMX ,
where M1 is the meridian of the torus V1, equipped with q, p.
Dehn surgery is naturally generalized to the case where y = Y1 U ... U Yr C M is a disjoint union (link) of equipped knots. The resulting manifold MY in this case is called the manifold obtained from the manifold M3 by Dehn surgery along the equipped link y. A link y = Y1 U ... U UYr C M is called trivial if knots y1, ..., Yr bound pairwise disjoint 2-discs d1, ..., dr C M.
Proposition 5 ([7]). Let y = Y1 U ... U Yr C M be a trivial link equipped with qx, p1, ..., qr, pr. Then
mY = M #L # ...Lp^qr.
2.3. Seifert fiber space
A solid torus V split into fibers of the form {x} x S1 is called a trivially foliated solid torus. Consider the solid torus V = B2 x S1 as the cylinder B2 x [0, 1] with the bases glued due to the angle rotation for coprime integers a, u, a > 1. The partition of the cylinder into segments of the form {x} x [0, 1] determines the partition of this solid torus into circles called fibers. The segment {0} x [0, 1] generates a fiber which we call exceptional, all other (ordinary) fibers of the solid torus wrap a times around the exceptional fiber and v times around the solid torus meridian. The number a is called the multiplicity of the exceptional fiber. A solid torus with such a partition into fibers is called a nontrivially fibered solid torus with orbital invariants (a, v).
A Seifert manifold is a compact, orientable 3-manifold M decomposed into disjoint simple closed curves (fibers) in such a way that every fiber has a neighborhood consisting of fibers, fiberwise homeomorphic to a foliated solid torus. Such a partition is called Seifert fibration. The fibers which correspond to the exceptional fiber under such homeomorphism of a nontrivially foliated solid torus are called exceptional.
Two Seifert fibrations M, M' are called isomorphic if there exists a homeomorphism h: M — — M' such that the image of each fiber of one bundle is a fiber of the second bundle. It is easy to show (see, for example, [8, Proposition 10.1]) that two bundles of a solid torus with orbital invariants (a1, v1); (a2, v2) are isomorphic (preserving the orientation of fibers) if and only if a1 = a2 (= a); v1 = v2 (mod a).
The base of a Seifert manifold M is a compact surface £ = M/~, where ~ is an equivalence relation such that x ~ y if and only if x and y belong to the same fiber. It is easy to show (see, for example, [8, Proposition 10.2]) that the base of any solid torus bundle is a disc. The base of any Seifert manifold is a compact surface and Seifert bundles with nonhomeomorphic bases are not isomorphic (see, for example, [8]).
Thus, any Seifert fibering M with a given base £ and orbital invariants (a1, v1), ..., (ar, vr),
r
r G N is obtained from the manifold £ x S1 by Dehn surgery along the link y = [J , where Yi =
i=1
= {sj x S1, si G £ is a knot with equipment fi, ai, vifi = 1 (mod ai). Therefore, the conventional notation for such a Seifert fibration is
M(£, ft), ..., K, Pr)).
Proposition 6 ([8, 9]). Seifert fibrations M (£, (a1,^1),..., (ar, /r )) and M' (£', (ai, /'1 ), ..., (a'rt, (i'ri)) are isomorphic if and only if there exists ô = ±1 such that :
• £ is homeomorphic to £';
• r = r'; ai = ai; /i = (mod a) for i g{1, ..., r};
r 0 r 0'
• if the surface £ is closed, then '¿r = à
i=i i i=i ai
Proposition 7 ([9, Proposition 1.12]). All closed orientable Seifert manifolds are prime except M (S2, (2, 1), (2, 1), (2, 1), (2, 1)) ^ RP3#RP3.
Proposition 8 ([10]). A 3-manifold admits a Seifert fibration with sphere base and at most two singular fibers if and only if it is homeomorphic to a lens space, so that
the only manifold which admits fibering without singular fibers is S2 x S1; M (S2, (a, 0)) - Lfiaa;
M (S2, (ai, 01), (a2, 02)) = Lpq, where p = 0a - a0, q = 0V - a^2 and ^02 -- a2^2 = 1
It follows from the above statement, in particular, that any lens space admits more than one Seifert fibrations. However, as the result below shows, any such fibration with base sphere cannot have more than two exceptional fibers.
Proposition 9 ([8]). Any lens space does not admit a Seifert fibration with base homeo-morphic to sphere and more than two exceptional fibers.
3. Dynamics of the flows G3 (M3)
This section is devoted to the proof of Lemma 1: the nonwandering set of any flow f1 e e G- (M3) consists of exactly three periodic orbits S, A, R, saddle, attracting and repelling, respectively.
Proof. The basis of the proof is the following representation of the ambient manifold M3 of the NMS flow f1 with the set of periodic orbits Perft (see, for example, [11])
M3 = [J OePerft WO = (J WO, (3.1)
aePer11
as well as the asymptotic behavior of invariant manifolds
el (WO) \ WO = U Wu,
OEPer t: Wgnw,¿=0
el (WO) \ WO = U .
OEPer t: W¿nwg=0
In particular, it follows from the above relations that any NMS flow has at least one attracting orbit and at least one repelling one. Moreover, if an NMS flow has a saddle periodic orbit, then the basin of any attracting orbit has a nonempty intersection with an unstable manifold of at least one saddle orbit (see [12, Proposition 2.3]) and a similar situation with the basin of a repelling orbit. ( )
Now let ff e G- (M3) and S be its only saddle orbit. It follows from the relation (3.1) that WSU \ S intersects only basins of attracting orbits. Since the set WSU \ S is connected and the basins of attracting orbits are open, WSU intersects exactly one such basin. Denote by A the corresponding attracting orbit. Since there is only one saddle orbit, there is only one attracting orbit. Similar reasoning for WSS leads to the existence of a unique repelling orbit R. □
4. Topology of ambient manifolds of flows of class G3 (M3)
In this section, we prove Theorem 1: flows of class G- (M3) admit all lens spaces Lpq, all connected sums of the form Lp, #RP3 and all Seifert manifolds of the form M (S2, (2, 1), (a1} ß1), (a2, ß2)). Namely, let the flow f G G- (M3) have the invariant Cft = (l1, m1, l2, m2). Then
1) if l1 = 0 and l2 = 0, then M3 is homeomorphic to the manifold L^
2) if l1 = 0 and l2 = 0, then M3 is homeomorphic to the manifold Ll m #RP3;
3) if l1 = 0 and l2 = 0, then M3 is homeomorphic to S2 x S1^w3-
4) if \l1\ = 1 and \l2\ > 1, then M3 is homeomorphic to the lens space Lpaq, where p = 202 — l2b, q = ^til _ p2TU2 = i (mod l2),b = 2 (mod 2);
5) if \l2\ = 1 and \l1 \ > 1, then M3 is homeomorphic to the lens space Lpaq, where p = 201 — l1b, q = ^til _ piTni = i (mod l{),b = 2 (mod 2);
6) if \l1l2\ = 1, then M3 is homeomorphic to the lens space Lb 2, b = 1 (mod 2);
7) if 111\ > 1 and \l2\ > 1 tl)en M3 is homeomorphic to the prime Seifert manifold M (S2, (2, 1), (l1, 01), (l2, 02)), 0imi = 1 (mod lj, i = 1, 2 and is not homeomorphic to any lens space.
Proof. The idea of the proof is to recognize that the sphere S3 is obtained by Dehn surgery along a link consisting of a saddle orbit S of the flow f1 and a knot 7 from the ambient manifold M3. Then, due to the relation (2.1), we have M3 = S3 _, which allows us to describe
' v n SUj
the topology of the manifold M3 using the set Cft = (l1, m1, l2, m2). Let us break down the discussion into steps.
1. Dehn surgery along a saddle orbit S. Let us show that the following relation is true for a saddle orbit S:
M3 = Lr!S.
Let us put
V+ = {(d1, d2, s) e V \ d1 ^ 0}, T+ = {(d1, d2, s) e dV \ d1 ^ 0}, V_ = {(d1, d2, s) e V \ d1 < 0}, T_ = {(d1, d2, s) e dV \ d1 < 0}.
Let h: dV ^ dVS be a homeomorphism such that
h(T+)= TSU, h(T_ )= TS.
Then h*((1, 0)) = (2, 1), which implies
K =(^b ^, b,c e Z.
Consider the Dehn surgery M'3 on M3 along the knot S with a neighborhood Vs and equipment b, c. Let vS: (M3 \ int V^ U V ^ MS be the natural projection. For simplicity, we keep
the notation of all objects on vS (M3 \ int Vs) the same as it was on M3 \ int VS and set S = = vS ({0} x S1), V~ = vS(V). Then M33 is the union of two solid tori VA = VA U vS(V+)
and VR = VR U vS(V-) such that Va n VR = dVa n dSR and hence M| = Lr s for some coprime integers a, b.
2. Reverse Dehn surgery on lens Lr s along the knot S. Let Ta = dsa, Tr = dVR and Lr s = = Va U sr. From Proposition 3 we find that M3 = (Lr s) where S is a knot with equipment -b, 2. For knots 5 cTa (= TR) denote by (5) a , (5)R the homotopy types of the knot 5 on tori Ta, TR, respectively. Then for cases 1)-3) from the definition of Cft we have the following relations.
1. If l1 = 0 and l2 = 0, then either (S)A = (0, 0) or (S)A = (0, 1). In the first case (S)R = = (0, 0) and (Ma)r = (l2, m2), which means r = l2, s = m2. Then by Proposition 4, S^ = Ll2 m2, where Ma is the meridian of the torus Ta equipped with m2, l2. Thus, M3 = S3 _ . Since the knots S U MA form a trivial link on sphere S3 (MA can be chosen
SUMA A v A
not to intersect S), by virtue of Propositions 5,
M 3 = SLMA = Lh, m2 #L2 ,1 = K m2 #RP3-
Similarly, if (S)a = (0, 1), then (Ma)r = (S)R = (l2, m2) and so r = l2, s = m2. Since Ma can also be chosen to be disjoint from S, it follows that
M3 = Lh ^^ #RP3.
2. If l1 = 0 and l2 = 0, then (S)R = (0, 1). Then (MR)A = (S)A = (l1, m1) and, hence, r = l1, s = m1, whence, from arguments similar to the above, we obtain
M3 = Lh ,n
3. If l1 = l2 = 0, then (S)R = (0, 1). Then (MR)A = (S)A = (0, 1) and, hence, r = 0, s = 1, whence it follows that
M3 = RP3#L0 ^ = RP3#S2 x S1.
3. Seifert fibration on manifold M3. To prove the remaining points, we note that in the case when l1l2 = 0, the manifold M3 = Va U Vs UVr has a Seifert fibration. Indeed, in this case, the fibration VS of the solid torus with exceptional fiber S and orbital invariants (2, 1) contains the knots KSU and KsS as fibers. This fibration extends to a solid torus Va and VR fibration with fibers A and R (which may or may not be exceptional), respectively, and with orbital invariants (l1, m1) and (l2, m2). In this way,
M3 = M (E, (2, b), (l1,ft), (l2,&)), Pi mi = 1 (mod y, b = 1 (mod 2).
Let us show that the base E of such a bundle is a 2-sphere.
Let ~ be an equivalence relation whose equivalence classes are the fibers of this fibration. Figure 2 shows the meridian disks Da, Ds, DR of the tori Va, Vs, VR, respectively, the segments containing equivalent points are shown in the same color. Gluing the equivalent points in the disks Da, Ds, DR, respectively, we obtain the disks Da = Va/~, Ds = VS, DR = VR, in which
DAnDs
Fig. 2. Disks Da, Ds, Dr
DRHDS
Fig. 3. Disks DA, DS, Dr
Fig. 4. E = S2
each fiber, except for the boundary fibers, is represented by one point and each boundary fiber is represented by two points on different disks (see Fig. 3). By gluing the equivalent points in the disks DA, DS, DR we obtain the sphere S2 (see Fig. 4), which is the base of the fibration given on M3. So,
M3 = M (S2, (2,6), (¡2 ,&)), Pi mi = 1 (mod (4.1)
1. If \l11 = 1, \l2\ > 1, then the fiber A is ordinary and, by Proposition 8,
M3 = M (S2, (2, b), (¡2, P2)) = Lp,q, where p = 2j32 — l2à, q = — fi2.
2. If \l1 \ > 1, \l2\ = 1, then the fiber R is ordinary and, according to Proposition 8,
M 3 = Lp q q,
where p = 2/3l — q = — /31.
3. If \l1l2\ = 1, then both fibers A, R are ordinary and, hence, by Proposition 8 and Proposition 2,
M3 = M (S2, (2, b)) = Lb22.
4. If \l1 \ > 1, \l2\ > 1, then M3 is a Seifert manifold with three exceptional fibers
M3 = M (S2, (2, b), (li, ßi), (l2, ß2)) = M (S2, (2, 1), (li, ßi ), (l2, ß2)),
o/ w j ^ ß1 ß2 b ßi ß2 1
ftm, = 1 (mod y, 7i + 7i + ^ = yi + r + ^-l1 l2 2 l1 l2 2
By Proposition 7, M3 is prime and, by Proposition 9, it is not homeomorphic to a lens space. □
Conflict of interest
The authors declare that they have no conflict of interest.
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