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4Динамические системы, 2020, том 10(38), №2, 129-139 MSC 2010: 37D05, 37D15
On Realization of Gradient-like Flows on the Four-dimensional Projective-like Manifold1
E. Gurevich, A. Chernov, A. Ivanov
National Reseasrch University Higher School of Economics Nizhnii Novgorod, 603155.
E-mail: egurevich@hse.ru, mrandche@gmail.com, artynn98@gmail.com
Abstract. In 1962 Eells and N. Kuiper provided manifolds admitting the Morse function with exactly three critical points. They shown that the dimension n of such manifolds takes the values 2,4, 8 and 16, and the critical points of the Morse function have indices 0, n/2 and n. Later these manifolds were called projective-like. In 2013 E. V. Zhuzhoma and V.S.Medvedev obtained a topological classification of gradient flows of such Morse function. In particular, they proved that all such flows on four-dimensional manifolds are topologically equivalent that means that there is only one projective-like manifold of dimension four (that is not true for higher dimension). In this paper, we study the relationship between the numbers of equilibrium states of various indices of a gradient-like flow on the projective-like manifold of dimension four. We also provide an algorithm of realization such flows with the given numbers of equilibrium states of different indices.
Keywords: gradient-like flow, heteroclinic curves, topological classification, projective-like manifolds.
1. Introduction and Statement of Results
Let Mn be a smooth closed connected manifold of dimension n. Recall that a flow ff on Mn is called Morse-Smale if its non-wandering set Qft belongs to a finite set of hyperbolic equilibrium states and closed trajectories, and invariant manifolds of different equilibrium states and closed trajectories have only transversal intersection. A Morse-Smale flow without closed trajectories is called gradient-like. S. Smale in [1] showed that for an arbitrary manifold Mn there exists a Morse function (a smooth function whose critical points are non-generated) defined on Mn, and it is possible to choose a metric on Mn such that the gradient flow of the Morse function will be a gradient-like flow. Hence, gradient-like flows exist on all manifolds.
xThis work was supported by the Russian Science Foundation under grant 17-11-01041, except the proof of Theorem 1 which was performed with support of the Laboratory of Dynamical Systems and Applications NRU HSE, of the Ministry of science and higher education of the RF grant ag. № 075-15-2019-1931.
© E. GUREVICH, A. CHERNOV, A. IVANOV
Recall that the sets
Wf = {q G Mn: lim f t(q) — p}, Wu = {q G Mn: lim f -t(q) — p}
p t^ + CX p t^+CX
are called stable and unstable manifolds of an equilibrium state p correspondingly.
According to [2, Theorem 2.3], if there is a gradient-like flow ft on a manifold Mn then Mn is a disjoint union of stable manifolds of all points from Qft and for any point p G Qft its stable and unstable manifolds are smoothly embedded open balls. Dimension dimWU of the unstable manifold of the point p is called a Morse index of p. It follows from hyperbolicity of the point p that dimWU G {0,1,...,n} and dim Wf + dim W^ = n. An equilibrium p such that dim W^ = 0 (dim W^ = n) is called a sink (a source), and an equilibrium p such that dimW^ G (0,n) is called a saddle point.
It follows from the observation above that for any gradient-like flow ft the set Qft contains at least one source and one sink. If the set Qft is exhausted by these two points, then the ambient manifold Mn is a sphere, and all such flows are topologically equivalent. According to [9] any gradient-like flow has an energy function — a Morse function decreasing along non-singular trajectories of ft such that the set of critical points of f coincides with the set Qft. Then the question of an existing of gradient-like flows with non-wandering set consisting of exactly three equilibrium states is reduced to the problem of existing of Morse function with exactly three critical points. Manifolds admitting such Morse function were studied in [7]. In particular, there was proven that the dimension of these manifolds takes the values n G {2, 4, 8,16} and the indices of the critical points equal 0, n,n. For n =2 this manifold is the projective plane.
Gradient-like flows with non-wandering set consisting of exactly three points were studied in [3], [4]. In these papers manifolds admitting such flows were called projective-like manifolds. It was also proved that for n = 4 all flows on a projective-like manifold which non-wandering set consists of exactly three hyperbolic equilibrium states are topologically equivalent. Hence, all four-dimensional projective-like manifolds are homeomorphic. This fact is not true in case n > 4, since, due to [7], in each dimension 8,16 there exist projective-like manifolds with different homotopy types.
In this paper, we do the first step to solution of a problem of topological classification of gradient-like flows on projective-like manifolds with arbitrary number of equilibria. Namely, we study a structure of a non-wandering set of gradient-like flows on a projective-like manifold of dimension four and provide an algorithm of a realization of such flows for given number of equilibria of different Morse indices.
For gradient-like flow ft on a four-dimensional manifold denote by lft the number of sink and source equilibrium states, by hft — the number of saddle equilibrium states of
Morse index two, and by kft the number of sаddle equilibrium stаtes of Morse indices one and three.
Main results of the paper are following.
Theorem 1. Let f be a gradient-like flow on the four-dimensional projective-Uke manifold M4. Then lf t-kf t +hf t = 3. If for any two different saddle equilibria p,q E Qf t the intersection Ws H W^ is empty then hft = 1.
Theorem 2. Let l > 2, k = 0, h > 1 be integers such that l — k + h = 3. Then there is a gradient-like flow f on the four-dimensional projective-like manifold such that lft = l, kft = k, hft = h.
2. The Structure of non-wandering set of gradient-like flows on four-dimensional projective-like manifolds
This section is devoted to the proof of Theorem 1. 2.1. Auxiliary results
Let us recall that a sphere Sk is the manifold homeomorphic to the standard sphere Sk = {(x1,... ,xk+i) C Rk+1| xf + • • • + x2k+1 = 1}, a ball (an open ball) Bn is the manifold homeomorphic to the standard ball (the interior of the standard ball) Bn = {(xi,...,xn) C Rn| xi + ••• + xkn < 1}.
The sphere Sk topologically embedded in a topological manifold Mn (1 < k < n—1) is called locally flat if for any point z E £k there exists a neighborhood Uz C Mn and a homeomorphism tpz: Uz ^ Rn such that tpz (Sk H Uz) = Rk C Rn. If the sphere Sk is not flat at a point z, then the point z is called the point of wildness and the sphere Xk is called wild.
The statement below follows from [2, Theorem 2.3].
Statement 1. Let ff be a gradient-like flow on a closed manifold Mn. Then
1. Mn = U w* = U Wp';
p&Qft p&Qft
2. for any point p E Qft the manifold W^ is a smooth submanifold of Mn;
3. for any point p E Qft and any connected component lu of set Wu \ p the closure
"p
cl of lu satisfy the equality cl \ U p) = (J W^.
qeQf -.Wfrn^Q
Item 1 of the Statement 1 and the fact that an unstable manifold of a hyperbolic equilibrium state p is a ball of dimension indp E {0,..., 4} lead to the fact that the set Qft of any gradient-like flow f1 contains at least one source and one sink. Indeed, in the absence of sinks (or sources), a manifold Mn of dimension n would be represented as a finite union of smoothly embedded balls of smaller dimension that is impossible.
Everywhere below we suppose that f is a gradient-like flow on projective-like manifold M4.
Denote by Qft the set of all equilibrium states of the flow ff which have the dimension of the unstable manifold equal to i E {0,1, 2, 3,4} and by |Qft | the capacity of the set |Qft|. Put lft = |Qf| + |Qft|, kft = |Qf | + |Qf |, and hft = |Qf |. It follows from [7] that Euler characteristic x(M4) of M4 is 3. Then due to Poincare-Hopf Theorem we have
lft — hft + kft = 3. (2.1)
It immediately follows from Equation (2.1) that if the set Qjt U Qft is empty then the set Qft consists of exactly three equilibrium states: a source, a sink, and a saddle with a Morse index two.
Let p,q E Qft are saddle points such that fl W^ = 0. Then the intersection Wp f Wqf is called heteroclinic intersection.
Lemma 1. Let a flow f has no heteroclinic intersections, and p E Qjt (p E Qf )• Then the closure clW£ (cl W^) of stable (unstable) manifolds W£ (W^) of the point p is a locally flat sphere of dimension 3 that divides the manifold M4 into two connected components•
Proof• Assume that the set Qft is non-empty and prove the lemma for an arbitrary point p E Qjt (the proof for the point p E Qf"1 is carried out similarly). It follows from item 3 of Statement 1 that for any point p E Q jt the closure cl of its stable manifold W ^ is the union of the manifold W ^ itself and a source equilibrium state ap. Therefore clW^ is a sphere of dimension (n — 1). Due to item 2 of Statement 1 the sphere cl W£ is smooth (and, therefore it is locally flat) at all points of W£. According to [8, St 3A.6] a sphere Sn-1 embedded in a manifold Mn of dimension n > 4 is either locally flat at each point or has more than a countable number of wildness points2. Hence, cl W ^ is a locally flat sphere.
Let us show that the sphere cl divides the manifold M4 into two connected components. Since, by virtue of [7], the fundamental group n1(M4) is trivial then M4
2In the paper [8] it is noted that this statement is a consequence of results of A. V. Chernavsky
and R. Kirby obtained independently in 1968. Earlier, in 1963, J. Cantrell proved the following: if the
sphere Sn-1 c Sn, n > 4, is wild and B is a set of points such that Sn-1 is locally flat in each point of the set Sn-1 \ B, then the set B consists of more than one point (see [6]).
is orientable. By [5, Theorem 3] a locally flat sphere Sn-1 in an orientable manifold Mn (n > 3) is cylindrically embedded, which means that there is a closed neighborhood V C Mn of a sphere Sn-1 and a homeomorphism h : Sn-1 x [—1,1] ^ V such that h(Sn-1 x {0}) = Sn-1. Therefore there is a neighborhood Vp of the sphere clW^, which is divided by the sphere cl into two connected components. Choose points x,y that belong to different connected components Vp \ cl and connect them with a smooth arc lp C Vp that intersects the sphere cl W ^ at the only one point. If cl W ^ does not divide M4, then there is an arc bp C M4 \ cl connecting the points x,y. By construction, the intersection index of the arc Xp = lp U bp and the sphere cl is 1 or —1 (depending on the choice of orientations). On the other hand, since nn-1(M4) is trivial, it is not difficult to choose a sphere Sn-1 C M4 \ \p, homotopic to the sphere
clWs. Since the intersection index is a homotopy invariant, the intersection index of the sphere Sn-1 and the arc Xp must be equal ±1, but since Sn-1 fl Xp = 0, it equals to zero. This contradiction proves that the sphere cl Ws divides the manifold M4 into two connected components. □
Remind that the set A is called an attractor of a flow ff if there is a closed neighborhood (a trapping neighborhood) V C Mn such that all trajectories of the flow ff intersect its boundary dV transversally, and A = P| ff (V). The set R is called
a repeller of the flow ft if it is an attractor for the flow f Set
t>0 -t
Af t = U KRft = U W
pen 01 ufii t penft unf un^
Lemma 2. If ft has no heteroclinic intersections then the set Aft is a connected attractor with a trapping neighborhood diffeomorphic to the ball.
Proof. It follows from [1, 9] that there is a Morse function p: M4 ^ [0, 4] such that the set of critical points of p coincides with the set Qf t, p(p) = ind(p) for any p E Qf t, and p(ff(x)) < p(x) for any point x E Q(ff) and t > 0. Let us show that the set V = p_1([0; 1, 5]) is a trapping neighborhood for Aft.
It follows from the definition that Aft C V. Since Aft is invariant then Aft C
fl f *(V). Let us prove that Aft = P| f l(V). Assume the opposite. Then there is a point
t>0 t>0
x E P| f t(V) \ Aft. Statement 1 implies that there is an equilibrium state p E Qft such t>0
that x E W^. Since the set P| ft(V) is closed and invariant then p E P| f t(V) \ Aft,
t>0 t>0 which is impossible, since the set V does not contain equilibrium states other than
those which belong to Aft. Therefore, Aft = P| fl(V) and Aft is an attractor.
t> 0
Let us prove that the trapping neighborhood V is connected. Then Af will be connected as the intersection of connected compact nested sets. Assume that V is disconnected, that is it can be represented as a union of two disjoint non-empty
invariant subsets E1,E2. Then the union (J = (J ft(E1 U E2) is disconnected.
peAft teR
Due to Statement 1, M4 = (J W s U Rft, then M4 \ Rft = (J Wps, so Mn \ Rft is
peAft peAft
disconnected. On the other hand, since the dimension of the set Rft does not greater than one, then Rft does not divide M4, therefore the set M4 \ Rft is connected. This contradiction proves that V and Aft are connected.
To prove that V is a ball let us prove that Aft does not contain subsets homeomorphic to a circle. Assume the opposite: let c c Aft be a simple closed curve. It follows from Items 1,3 of Statement 1 that the set Aft \ Q1 is a finite set of arcs lying in the disjoint union of stable manifolds of sink equilibria. Therefore there is an equilibrium state p E Q^t such that p E c. Due to Lemma 2, the set cl W ^ divides the ambient manifold M4 in two connected components. Therefore cl W ^ also divides the curve c, so there is a point x E c f cl different from p. The point x cannot be a source, since Aft does not contain sources by construction. The point x cannot be a sink or since x E W ^ \ p and only non-wandering point in W ^ is p. Hence, x belongs to a one-dimensional unstable manifold of some point q E Q^t, but we supposed that ft has no heteroclinic intersections, so we get a contradiction.
Thus the set Aft can be represented as a connected graph without cycles, whose vertices are sink points and edges are one-dimensional unstable manifolds of saddle points. Then |Q011 = Q11 + 1. It follows from Morse theory that the set V is a smooth subset of M4 obtained from the disjoint union of |Q011 balls by gluing |Q111 handles of index 1. Using induction one can easy prove that V is a ball. □
Set lift = (J Wp. Considering f-t and applying the Lemma 2 one can get
p'
p—p
^ (\лг\л-лг\г\ "I "f" TO <-\ V/ЛГ» 1 nr -P/-"\Y>
Ufif
that Rft is a connected attractor for f-t (hence, it is a repeller for ft) with a trapping neighborhood W diffeomorphic to the ball. This observation and Lemma 2 immediately lead to the following statement.
Corollary 1. In assumption of Lemma 2 there are smoothly embedded balls V,W c M4 such that:
1. Aft C V, Rft C W;
2• trajectories of the flow ft are transversal to the boundaries of the balls V, W and are oriented out of the interior of W to the interior of V;
3. the non-wandering set of the flow ft restricted on the set M4 \ (V U W) consists of the equilibrium states of index 2.
2.2. Proof of Theorem 1
Remind that a connected sum of smooth orientable connected manifolds M'n,M'^ is the manifold M^M? obtained as follows. Let B? C Mn, BC Mn be two balls. Then the manifold Mi^^M^ is the result of gluing manifolds Mi1 \ B'DM'Z \ B'n by a reversing the natural orientation diffeomorphism h: dBn — dBn. According to [10, Lemm 2.1], the connected sum operation is defined the unique (up to diffeomorphism) manifold and does not depend on the choice of balls and a gluing homeomorphism.
Proof of Theorem 1. Suppose that a gradient-like flow ft on the projective-like manifold M4 has no heteroclinic intersections. Let us show that the set of saddle equilibrium states of the flow ft contains exactly one equilibrium state whose Morse index equals two. Then the equality lft — kft = 2 will immediately follow from Hopf-Poincare formula 2.1.
Let W,V C M4 be balls described in Corollary 1. Then M4 \ int (W U V) is the manifold with boundary consisting of two (n — 1)-spheres. Let B+, B™ be two standart balls enriched by vector fields correspondingly. Glue balls
B+, Bn to M4 \ int W U V) with reversing the natural orientation diffeomorphism p: dB+ U dB™ — dW U dV, denote by M4 the resulting manifold and by : B+ U B™ U M4 \ int (W U V) — M4 the natural projection. It is possible to choose the diffeomorphism p in such a way that it induce on M4 a gradient-like flow ft such that flM4\int (wuv) = n f11 m 4\int (wuv) and the restrictions f^ J^b™ are topologically equivalent to dilatation and contraction correspondingly. So, non-wandering set of the flow ft consists exectly of one source, one sink, and |Qft | saddles of index 2. The operation of gluing balls is equivalent to taking a connected sum with two spheres, so the manifold M4 is diffeomorphic to the original manifold M4. Then, due to Poincare-Hopf formula 2.1, lQk11 = 1. The Theorem 1 is proven.
3. Realization of gradient-like flows on four-dimensional projective-like manifolds
This section is devoted to the proof of the Theorem 2. Let l > 2, k > 0 and h > 1 be integers such that l — k + h = 3.
We are going to construct a gradient-like flow ft such that the number lft of sink and source equilibrium states of ft equals l, the number hft of saddle equilibrium states of Morse index two equals h, and the number kft of saddle equilibrium states of Morse index different from two equals k.
To construct the desired flow we define below auxiliary flows gt1, g2 on the projective-like manifold M4 and the sphere S4, respectively, with the following properties:
1. the non-wandering set of the flows g\ consists exactly of one source, (k — h +1) saddles of Morse index one, one saddle of Morse index two and (k — h + 2) sinks;
2. the non-wandering set of the flows g1 consists exactly of one sink, one source, (h — 1) saddles of Morse index one and (h — 1) saddles of Morse index two.
Choose the balls Bn C M4, Bin C S4 that intersect with the sets Qgt, Qgt exactly at one point: the sink and source respectively, lying in the interior of the balls B'[,B'^. We form a connected sum of manifolds M4,S4 by cutting out the interiors of the balls and gluing the resulting manifolds by a diffeomorphism to induce on the
manifold M4$S4 a gradient-like flow ft such that the non-wandering set of the flows ft consists exactly of l = 3 + k — h sinks and sources, k saddles of the index 1, and h saddle of the index 2 (see, for example [14]). The connected sum operation with a sphere does not change the topological type of the manifold, so the manifold M4$S4 is the projective-like manifold, so ft is the desired flow.
3.1. Construction of the flow g\
Let us describe the buiding of the flow g t step by step.
Step 1. Realization of a gradient-like flow g0 whose nonwandering set consists of exactly three equilibria: a source, a sink and a saddle of Morse index two.
Let us define the flow fk on the handle H\ = Bk x B4-k of the index k E {0,..., 4} by the following system of differential equations
x = x,x E Bk y = —y,y E B4-k.
A non-wandering set of the flow fk consists of a single equilibrium state O which Morse index is k. For k > 0 trajectories of the flow fk having non-empty intersection
k4
with the foot F4 = 5Bk x B4 k of H4 intersect the foot transversally and directed
outside of H4.
First, we are going to obtain a projective-like manifold M4 by sequentially gluing to the handle H0 the handles H| and H%. After the gluing handles, the flows f0, f2,, f\ will induce on M4 the desired flow g0.
The foot Ff = S1 x B2 is a solid torus whose core S1 x {O} (here O — the center of the ball B2) belongs to the unstable separatrix of the saddle equilibrium state of the
flow ft. Remark that dH4 = S3. Let c E S3 be a node (a simple closed curve), Nc is its closed neighborhood, and Pc = S3 \ int N.
Let us denote by a manifold with a boundary obtained by gluing the handle H4 to the handle by means a diffeomorphism p: F^ — Nc. We are going to glue the handle H44 to X and obtain a closed manifold, then the boundary of X must be diffeomorphic to the sphere S3. For this purpose we should choose the gluing diffeomorphism p: — dH4 and the node c.
As the gluing diffeomorphism p: — Nc is a solid torus diffeomorphism, it maps the meridian of F^ to the meridian of Nc. But the meridian of F2 is the longitude of solid torus dH% \ int F%. So, the gluing operation is a nontrivial surgery. By virtue of [11, Theorem 1], no nontrivial surgery along a nontrivial node will give a sphere. It follows that the knot c must be the boundary of a 2-disk in S3. Hence, Pc is the solid torus. Let p send the longitude of Nc to the curve of homotopy type (1,1) in dNc. Then, due to [12], dX will be the sphere.
Now we are able to glue the handle Hj to X by an arbitrary orientation reversing diffeomorphism ^: dH% — dX. As a result, we get a closed manifold M4 carrying a gradient-like flow g\ whose non-wandering set consist of exactly three equilibrium states. Hence, M4 is the projective-like manifold.
Step 2. A realization of a gradient-like flow ht on the sphere S4 whose non-wandering set consists of exactly one source, k saddles of index 1, and k + 1 sink.
Define a gradient-like flow on the sphere S4, which has a non-wandering set consisting of exactly one source, k saddles of index 1, and k + 1 sinks.
We construct k copies of the sphere S4,... ,S%, each of which carries the flow i E {1,... ,k} whose non-wandering set consists of exactly one source ai, one saddle ai of index 1, and two sinks . To do this, we glue one handle of index 1 to two
handles of index 0 to get the ball carrying a gradient-like flow whose trajectories are transversal to the boundary of the ball and the non-wandering set consists of two sinks and one saddle. Then we glue the handle H44 to the obtained manifold. As a result, we get the desired flow
Select a ball B4 C S4 (B| C S%) that intersect the set Q^t (Q^t) exactly at one point which is the sink (the source ak) lying in the interior of the ball Bn(Bn). We define a connected sum of spheres S4, S% by cutting out the interiors of balls B4, B4 and gluing the resulting manifolds with the boundary by an orientation-inverting diffeomorphism h1kk: dBn — dBn such that h1tk(Wu) n WS = 0. The gluing operation induce a
2 ^^ lb1,2Vy a!> ' <72
gradient-like flow without heteroclinic intersection on the connected sum S^S4. Set S4,2 = S4№l. The non-wandering set of the flow 2 consists of one source, two saddles of index 1, and three sinks. Similarly, we form a connected sum of the spheres
S42 and S3j, and so on. After k steps, we get the desired flow ^t. Step 3. Construction of the desired flow g\
Let us consider the projective-like manifold M4 carrying the flows g0 defined on the Step 1 and the sphere S4 carrying the flow ht defined on the Step 2. As it described above, it is possible to construct the connected sum M4$S4 and induce the desired flow gt1 on the M4$S4.
3.2. Construction of the flow g2
Let us construct an auxiliary gradient-like flows rf on the sphere S4 whose non-wandering set consists exactly on one source, one sink, and two saddles of Morse index one and two respectively. In [13] it is proved that the intersection of invariant manifolds of these two saddles is non-empty and consists of finite number of non-compact curves (trajectories) that are called heteroclinic curves. Then we take (h — 1) > 1 copies of spheres with carrying such flows and construct the connected sum of the spheres as it described above. As a result we obtain the desired flow g2.
To construct a flow rf let us construct a mainfold M1 by gluing the hande H1 to the hand H0 by meanse of an arbitrary smooth embedding g: S0 x B4 ^ S3. Then d M1 is homeomorphic to S2 x S1 and flows f0, f induce on M1 a gradient-like flow n\ whose non-wandering set consists of exactly two equilibria: a source u and a saddle a1 of Morse index one.
Set S2t = W^1 fdM1. By construction S2 t is the 2-sphere which does not bounds any ball in dM1. Then there is a homeomorphism 9: S2 x S1 ^ dM1 such that 9(S2 x {x}) = S^t, x E S1. Set c = 9(z x S1), z E S2 and denote by Nc C d M1 a tubular neighborhood of the node c. Let j: S1 x B2 ^ Nc be a diffeomorphism such that ¡(S1 x {O}) = c. Denote by M2 a manifold obtained by gluing the handle H2 to M1 by means of j. The boundary of M2 is the result of gluing two solid tori dH2 \ int (S1 x B2) and dM1 \ int Nc by means of the diffeomorphism n|SixSi that sends a longitude of dH2 \ int (S1 x B2) to the meridian of the solid torus dN1. Hence dM2 is 3—sphere. More over, due to [15, Theorems 3.30., 3.34], the manifold M2 is diffeomorphic to the ball H0.
Glue M2 and the hand Hf to get the sphere S4 and the desired gradient-like flow nt.
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Получена 14.05.2020
