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This article is cited in 2 scientific papers (total in 2 papers) On classification of Morse–Smale flows on projective-like manifolds V. Z. Grines, E. Ya. Gurevich National Research University "Higher School of Economics"
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     § 1. IntroductionA flow ft on a closed smooth manifold Mn of dimension n⩾1 is called a Morse–Smale flow if its non-wandering set Ωft consists of finite number of hyperbolic equilibrium states and closed trajectories, and stable and unstable invariant manifolds of the equilibria and closed trajectories either do not intersect each other or intersect transversely. A Morse–Smale flow without closed trajectories is called a gradient-like flow. According to [1], [2], for any gradient-like flow there exists an energy function, that is a Morse function, which strictly decreases along non-singular trajectories of the flow. The set of critical points of the energy function coincides with the set of equilibrium states of the flow, while local minina (local maxima) correspond to sinks (sources), and saddle critical points correspond to the saddle equilibrium states. Moreover, the index of the critical point coincides with the dimension of the unstable invariant manifold of the corresponding equilibrium state.1 Recall that the number ind(p), equal to the dimension of the unstable manifold of the hyperbolic equilibrium state p, is called its Morse index. Since Morse function on any compact manifold has at least one local maximum and one local minimum, then for any gradient-like flow ft the set Ωft contains at least two equilibria: a source and a sink. If the set Ωft is exhausted by these two points, then it is not difficult to prove that the ambient manifold Mn is homeomorphic to the sphere, and all such flows are topologically equivalent (see, for example, [3], Theorem 2.2.1, where a similar fact for gradient-like cascades was proved, which can be easily adapted for flows). Examples of Morse functions with exactly three critical points (a minimum, a maximum, and a saddle) are constructed in [4], § 19.3, and in [5], § 2. It follows from the results below, that the real and complex projective planes are the only manifolds (up to homeomorphism) of dimension 2 and 4, correspondingly, admitting the Morse function with exactly three critical points. Consequently, these manifolds are the only manifolds of dimension 2 and 4, admitting gradient-like flows with exactly three equilibria. All such flows (on the same manifolds) are turned to be topologically equivalent, that is not true for similar flows on the manifolds of greater dimension. These results give motivation for obtaining the topological classification of gradient-like flows with an arbitrary non-wandering set on the complex projective plane. The general approach to topological classification of gradient-like flows is based on Poincare–Bendixson and Smale Theorems, that establish the possibility of choosing a finite number of invariant manifolds that cut an ambient manifold into domains with the same asymptotic behavior of trajectories. It seems natural that the mutual arrangement of such invariant manifolds can be described in combinatorial terms, and the isomorphism of the combinatorial invariants is a necessary condition for two flows to be equivalent. As papers of Leontovich, Mayer, Peixoto, Oshemkov, Sharko, Fleitas, Pilyugin, Umanskii show, the isomorphism of combinatorial invariants is not only a necessary but also a sufficient condition of topological equivalence in meaningful classes of gradient-like flows on manifolds. It should be emphasized that all known classification results for flows are obtained under the assumption that any saddle equilibrium state has either stable or unstable manifold of dimension 1. In this case, the existence of the energy function guarantees that the closures of invariant manifolds which do not participate in heteroclinic intersections, are locally flat (see Proposition 4). However, if the carrying manifold has dimension four and greater, and the set of saddle equilibria contains points with invariant manifolds of codimension two, the closures of these invariant manifolds may be wildly embedded, which makes impossible to obtain complete classification in combinatorial terms. Similar effect was discovered earlier for Morse–Smale cascades on three-dimensional manifolds in papers [6], [7]. In [7], [8] new topological invariants of such cascades was provided and complete topological classification was obtained. We show in Lemma 1 that the non-wandering set of any gradient-like flow without heteroclinic intersections on the complex projective plane contains exactly one saddle with two-dimensional invariant manifolds, moreover, the closures of these manifolds are locally flat spheres. Due to this fact, in the present work it is possible to obtain the complete topological classification of such flows in combinatorial terms. Eels and Kuiper [5] studied manifolds, admitting a Morse function with exactly three critical point. In particular, they obtain the following properties of such manifolds and Morse functions. Statement 1. Let Mn be a connected closed manifold of dimension n and φ:Mn→R be a Morse function, the set of critical points of which consists exactly of three points. Then 1) n⊂{2,4,8,16}; 2) critical points of φ have indices 0, n/2, n; 3) Mn is the disjoint union of the open ball of dimension n and the sphere of dimension n/2; 4) M2 is diffeomorphic to projective plane; 5) for n⩾4, Mn is simply connected and orientable; M4 is homotopy equivalent to the complex projective plane. For n=8 (16) there exist six (sixty) of homotopy types of such manifolds. Manifolds, admitting a Morse function with exactly three critical points, are called Eels–Kuiper manifolds. Zhuzhoma and Medvedev in [9], [10] showed that for n=4 all gradient-like flows whose non-wandering set consists of exactly three points, are topologically equivalent. This implies that for n=4 all Eels–Kuiper manifolds are homeomorphic to the complex projective plane. Moreover, due to [9], [10], the topology of Eels–Kuiper manifolds may be refined as follows. Definition 1. A manifold Mn is said to be projective-like if: 1) n⊂{2,4,8,16}; 2) Mn is the disjoint union of the open ball of dimension n and a locally flat sphere2 of dimension n/2. According to [10], Theorem 1, a manifold Mn admits a gradient-like flow ft, non-wandering set of which consists exactly of three points if and only if Mn is projective-like. Hence the following statement holds. Indeed, if Mn is projective-like, then it admits a gradient-like flow ft, non-wandering set of which consists of exactly three points. It follows from [1], [2] that the flow ft has an energy function – a Morse function, the set of critical points of which coincides with the set of equilibria of the flow ft. Then Mn is Eels–Kuiper manifold. On the other hand, if Mn is Eels–Kuiper manifold, then it admits a Morse function with exactly three critical points. A gradient flow of this function is gradient-like, hence, due to [10], Theorem 1, the manifold Mn is projective-like. We denote by G(M4) a class of gradient-like flows on the complex projective plane M4 such that for any ft∈G(M4) invariant manifolds of different saddle equilibria does not intersect each other. We provide necessary and sufficient conditions for flows in G(M4) to be topological equivalent, and describe an algorithm of realization of all classes of topological equivalence. We now denote by Ωift the set of all equilibria of the flow ft∈G(M4) with a Morse index i∈{0,1,2,3,4}. The following statement proved below in § 4 allows to reduce a problem of topological classification of the flows under consideration to a combinatorial problem. Lemma 1. For any ft∈G(M4) the following statements hold. 1) If p∈Ω1ft (p∈Ω3ft), then the closure clWsp (clWup) of the stable (unstable) manifold Wsp (Wup) of p is a locally flat sphere of dimension 3 that divide M4 into two connected components. 2) The set Ω2ft consists exactly of one equilibrium state, and the closures of its invariant manifolds are locally flat two-dimensional spheres. We are going to show that the classes of topological equivalence of flows in G(M4) are distinguished by means of bi-color graph, that is defined below similarly to [11]. Denote by Lft the set of all spheres {clWsp,p∈Ω1ft} and {clWuq,q∈Ω3ft}, and by kft the number of these spheres. Since, due to Lemma 1, each sphere from the set Lft divides M4 into two connected components, the set M4∖(⋃p∈Ω1ftclWsp∪⋃q∈Ωn−1ftclWuq) consists of mft=kft+1 connected components D1,…,Dmft. Denote by Dft the set of all these components. Definition 2. A bi-color graph of a flow ft∈G(M4) is a graph Γft, defined as follows (see fig. 1): 1) the set V(Γft) of the vertices of the graph Γft is isomorphic to the set Dft, the set E(Γft) of the edges of the graph Γft is isomorphic to the set Lft; 2) the vertices vi, vj are incident to an edge ei,j if and only if the corresponding domains Di,Dj have a common boundary component; 3) the edge ei,j has the color s (u) if it corresponds to a manifold clWsp∈Lft (clWuq∈Lft); 4) the graph Γft has a unique marked vertex v∗ corresponding to the domain D∗⊂Dft which contains the saddle equilibrium state whose Morse index equals 2. Definition 3. Graphs Γft,Γgt of flows ft,gt∈G(M4) are isomorphic if there exists an isomorphism ξ:Γft→Γgt preserving the colors of the edges and the marked vertex. Recall that a tree is a connected graph such that for any pair of its vertices there exists a single path connecting these vertices. We prove in § 6 that a bi-color graph of any flow ft∈G(M4) is the tree. The main results of the paper are the following. Theorem 1. Flows ft,gt∈G(M4) are topologically equivalent if and only if their bi-color graphs Γft, Γgt are isomorphic. Theorem 2. For any tree Γ with a marked vertex v∗∈V(Γ) whose edges are arbitrarily colored in two colors, there exists a flow ft∈G(M4) whose bi-color graph Γft is isomorphic to Γ. § 2. Auxiliary topological factsThis section contains auxiliary topological facts used in the paper. 2.1. Homotopy groups of the ambient manifoldThe following statement clarifies properties of the complex projective plane. Proposition 1. Let M4 be the complex projective plane. Then homotopy groups π1(M4), π3(M4), π4(M4) are trivial and groups π2(M4), π5(M4) are isomorphic to Z. We preliminarily give all the definitions and facts necessary for the proof. Let X, Y be topological spaces. Continuous maps h,g:X→Y are said to be homotopic if there exists a continuous map (a homotopy) H:X×[0,1]→Y such that H|X×{0}=h,H|X×{1}=g. A homotopy H:X×[0,1]→Y is often presented as a family ht:X→Y of maps such that H(x,t)=ht(x), x∈X, t∈[0,1]. Let Ii⊂Ri, i⩾1, be a unit cube, ∂Ii be its boundary, and let f:Ii→X be a continuous map such that f(∂Ii)=x0, where x0∈X is a point in X. Denote by [f] a class of maps homotopic to f such that every map and the homotopy maps ∂Ii to x0. A set of equivalent classes [f], [g] with the operation ∗ given by the rule is called an i-dimensional homotopy group and is denoted by πi(X,x0). A group π1(X,x0) is called a fundamental group.A bundle is a triple (E,p,B), where E is a topological space, called the total space, B is another topological space, called the base space of the bundle, and p:E→B is a surjective map called the projection. The map p itself is often called a bundle. For each element b∈B, the fibre over this element is defined as a subset Fb⊂E of all pre-images of the element b. A bundle (E,p,B) is called a fiber bundle if for any point b∈B there is a neighborhood Ub⊂B and a homeomorphism φ:p−1(Ub)→Ub×Fb such that p|p−1(Ub)=proj1φ|p−1(Ub), where proj1:Ub×Fb→Ub is a projection defined by proj1(x,y)=x, x∈Ub, y∈Fb. Let (E,p,B) be a fiber bundle with a fiber F and a connected base space. Fix a point b0∈B. The map p:X→B induces a homomorphism p∗:πi(E,e0)→πi(B,b0), where e0∈Fb0. The inclusion i:Fb→E induces a homomorphism i∗:πi(Fb,e0)→πi(E,e0). For i⩾1, we define a map ∂∗:πi(B,b0)→πi−1(Fb,e0), which is a homomorphism for i⩾2, in the following way (see. [4], Ch. 4, § 14.3). The map f:(Ii,∂Ii)→(B,b0) is a homotopy, connecting constant maps φ0,φ1:∂Ii→b0. According to Covering Homotopy Theorem, there exists a homotopy ˜φt:∂Ii→E such that ˜φ0(∂Ii)=e0 and p˜φt=φt. Since φ1(∂Ii)=b0, ˆφ1(∂Ii)⊂Fb0. We note that the set ∂Ii is homeomorphic to the cube Ii−1 with the boundary ∂Ii−1 contracted to a point. Then define the image ∂∗f of the map f as the homotopy class of the map ˆφ1:(Ii−1,∂Ii−1)→(F,e0). A sequence of the groups and homomorphisms is called an exact sequence if for any i the image Imφi={φi(g),g∈Gi} of the homomorphism φi coincides with the kernel Kerφi−1={g∈Gi−1:φi−1(g)=0} of the homomorphism φi−1.The proof of the following statement can be found, for example, in [4], Theorem 14.3. Recall that a sphere Sk of dimension k⩾0 is the manifold homeomorphic to the unit sphere A ball (an open ball) Bn of dimension n⩾1 is the manifold homeomorphic to the unit ball (the interior of the unit ball)A canonical complex plane is the set CP2 of all equivalence classes [a,b,c] of triples (a,b,c)∈C3∖(0,0,0) with respect to equivalence relation (a,b,c)∼(a′,b′,c′) if (a,b,c)=(λa′,λb′,λc′), where λ is a non-zero complex number. Consider the unit five-dimensional sphere as the subset S5={(a,b,c)∈C3∣|a|2+|b|2+|c|2=1} of the space C3 and identify points of the sphere in the equivalence relation (a,b,c)∼(λa′,λb′,λc′), |λ|=1. A quotient space S5/∼ under this equivalence relation if homeomorphic to CP2. According to [4], Ch. 4, § 14.3, the following statement holds. Statement 4. A canonical projection p:S5→CP2 is a fiber bundle with a fiber homeomorphic to the circle. We use the exact sequence of the fiber bundle p:S5→CP2 to prove Proposition 1. Proof of Proposition 1. Consider the following subsequence of the exact sequence of the fiber bundle p:S5→CP2: The group π0(S1) coincides with the number of connected components of S1 and, consequently, is trivial, hence Ker∂∗=π1(CP2). Since the sequence is exact, Ker∂∗=Imp∗, therefore π1(CP2)=Imp∗ and p∗ is an isomorphism. Since π1(S5) is trivial, π1(CP2) is also trivial.
Let us prove that π3(CP2),π4(CP2) are trivial. Consider exact subsequences and apply the similar arguments taking into account that the groups πi(S1) for i⩾2 and πj(S5) for 0⩽j<5 are trivial.Consider an exact subsequence Since π1(CP2) is trivial, the map ∂∗:π2(CP2)→π1(S1) is an isomorphism. Since π1(S1) is isomorphic to Z, π2(CP2) is also isomorphic to Z.Similarly, due to exactness of the sequence and the fact that π5(S5) is isomorphic to Z, one gets that π5(CP2) is isomorphic to Z. The Proposition is proved.2.2. Dehn SurgeryA solid torus is the manifold Π homeomorphic to the direct product S1×B2 of the circle and the two-dimensional disk. Let x∈S1, y∈∂B2 be arbitrary points and φ:S1×B2→Π be a homeomorphism. Curves are called the canonical meridian and longitude, respectively, of the solid torus Π.We say that curves μ,λ lying in the boundary torus are meridian and longitude if they are homotopic to m, l, correspondingly. Since the fundamental group π1(T) of the torus is isomorphic to the group Z×Z, the homotopy class [γ] of any oriented closed curve γ∈T is defined by a pair of co-prime integers (p,q). We assume that [l]=(1,0), [m]=(0,1).According to [13], Ch. 2, C, Theorem 4, the following statement holds. Statement 5. Let ψ:∂Π→∂Π be a homeomorphism. A homeomorphism Ψ:Π→Π such that Ψ|∂Π=ψ|∂Π exists if and only if ψ preserves the homotopy class of the meridian. Due to [13], Ch. 2, B, C, a homeomorphism ψ:T→T induces a homomorphism ψ∗:π1(T)→π1(T) which is uniquely determined by an unimodular integer matrix Aψ such that [ψ(γ)]=(p,q)Aψ for [γ]=(p,q). According to [13], Ch. 2, C, Theorem 4, the following statement holds. Corollary 1. A homeomorphism ψ:T→T is isotopic to the identity map id:T→T if and only if ψ preserves the homotopy classes of the meridian and longitude. Proof. Set
Then [ψ(1,0)]=(1,0)Aψ=(a,b), [ψ(0,1)]=(0,1)Aψ=(c,d). Hence the homeomorphism ψ preserves the homotopy classes of the meridian and longitude if and only if the matrix Aψ is unite. Hence, due to Statement 6, homeomorphism ψ is isotopic to identity. The Corollary is proved. Recall that a simple closed curve C⊂S3 is called a knot. A knot C⊂S3 is said to be trivial if there exists a homeomorphism φ:S3→S3 such that Let C⊂S3 be a knot and ΠC⊂S3 its tubular neighborhood. Dehn Surgery along C⊂S3 is a way of obtaining a new closed manifold by gluing two manifolds with boundary S3∖intΠC and S1×B2 by means of a homeomorphism φ:∂(S1×B2)→∂ΠC. Let m⊂∂(S1×B2) be a canonical meridian and [ϕ(m)]=(p,q). The ratio q/p is called the surgery coefficient. A surgery with zero coefficient is said to be trivial. It follows from the Statement 5 that trivial surgery does not change the topology of the manifold. By [14], Theorem 2, the following statement is true. 2.3. Local homeomorphisms extendingDue to [15], Ch. 8, Theorem 3.1, 3.2, the following statement holds. Statement 8. Let ei,e′i:Bn→intBn, i∈{1,…,k}, be smooth embeddings such that ei(Bn)∩ej(Bn)=∅, e′i(Bn)∩e′j(Bn)=∅ for any i,j∈{1,…,k}, i≠j. Then there exists a homeomorphism h:Bn→Bn such that 1) h|∂Bn=id; 2) hei=e′i, i∈{1,…,k}. Proposition 2. Let C⊂S3 be a trivial knot, Π0⊂S3 be its tubular neighborhood, and let e,e′:S1×B2→intΠ0 be topological embeddings such that e(S1×{x})=e′(S1×{x}), x∈∂B2. Then there exists a homeomorphism θ:S3→S3 such that 1) θ|S3∖intΠ0=id; 2) θe=e′. Proof. Since e, e′ are the embeddings of the solid tori, due to Statement 5 the homeomorphism e′−1e:S1×B2→S1×B2 maps the meridian of solid torus S1×B2 into a curve homotopic to it. It follows from the condition of the Proposition, that the homeomorphism e′−1e maps the homotopy class of longitude of S1×B2 into itself. Then, according to Corollary 1, there exists an isotopy such that
Set where φ:∂S1×B2→S1×S1×{0} is the identity homeomorphism.Define a homeomorphism H:Π→Π acting identically on the set S1×B2 and equal to for points (x,t) from the set S1×S1×[0,1].Set Π=e(B2×S1), Π′=e′(B2×S1). It follows from [16], Theorem 3.3, that manifolds Π0∖intΠ, Π0∖intΠ′ are homeomorphic to the direct product of the torus and the interval. Then there are two foliations {Tq}, {Ip} on Π0∖intΠ with leaves homeomorphic to the torus S1×S1 and the segment [0,1], respectively, such that connected components of the boundary of the manifold Π0∖intΠ belongs to the foliation {Tq} and for every pair of leaves Tq, Ip the intersection Tq∩Ip is non-empty and consists of a single point. Assume that the leaves of the foliation {Tq} are parametrized by points q∈[0,1], and the leaves of the foliation {Ip} are parametrized by points p∈∂Π=T0. Hence, any point from the set Π0∖intΠ is determined by two coordinates (p,q), p∈∂Π, q∈[0,1]. Define an embedding ˜e:S1×S1×[0,1]→Π0∖intΠ by The embedding e:B2×S1→Π may be extended up to the embedding E:Π→Π0 by Similarly, extend the embedding e′:B2×S1→Π′ up to the embedding E′:Π→Π0. Lastly, set The map θ:S3→S3 is the desired homeomorphism. The proposition is proved.Proposition 3. Let S2⊂S3 be a smoothly embedded two-dimensional sphere, A0⊂S3 be its tubular neighborhood, and let e,e′:S2×[0,1]→intA0 be topological embeddings. Then there exists a homeomorphism θ:S3→S3 such that 1) θ|S3∖intA0=id, 2) θe=e′. Proof. Set A=e(S2×[0,1]), A′=e′(S2×[0,1]). A homeomorphism e′−1e|S1×{0} is an orientation preserving homeomorphism of the sphere, hence it is isotopic to the identity map. It follows from the Annulus Theorem that each of the set A0∖intA, A0∖intA′ has two connected components homeomorphic to the direct product S2×[0,1]. Further constructions is similar to those from the proof of Proposition 2. The proposition is proved.
§ 3. Auxiliary facts characterizing the dynamics of gradient-like flowsIn this section, we present a number of important auxiliary facts that help to describe the dynamics of flows from the class under consideration in a canonical way. The following statement follows from [1], Theorem B, [2], Theorem 1. Statement 9. For any gradient-like flow ft on a closed manifold Mn there exists a Morse function φ:Mn→[0,n] such that 1) the set of the critical points of the function φ coincides with the set Ωft; 2) φ(ft(x))<φ(x) for any point x∉Ω(ft) and for any t>0; moreover, all non-closed trajectories of the flow intersect the level sets of the function φ transversally; 3) φ(p)=ind(p) for any p∈Ωft; 4) in a neighborhood of any point p∈Ωift there exist local coordinates3 y1,…,yn such that the function φ can be presented as φ=φ(p)−y1−⋯−y2i+y2i+1+⋯+y2n. The statement below follows from [18], Theorem 2.3. Statement 10. Let ft be a gradient-like flow on a closed manifold Mn, n⩾1. Then 1) Mn=⋃p∈ΩftWsp=⋃p∈ΩftWup; 2) for any point p∈Ωft the manifold Wup is a smooth submanifold of the manifold Mn; 3) for any point p∈Ωft and any connected component lup of the set Wup∖p the equality cllup∖(lup∪p)=⋃q∈Ωf:Wsq∩luq≠∅Wuq holds. From item 1) of the Statement 10 one can easily obtain the following fact. Corollary 2. The non-wandering set Ωft of any gradient-like flow ft contains at least one sink and one source. Proof. Assume the contrary. For definiteness, we assume that the set of source equilibrium states is empty. By Part 1) of Statement 10, the manifold Mn is represented as a union of unstable manifolds of equilibrium states. Since the equilibrium states of the flow ft are hyperbolic, the unstable manifold of the equilibrium state p is a ball of dimension k={0,…,n}. Thus, in the absence of sources, a manifold Mn of dimension n can be represented as a finite union of smoothly embedded balls of dimension less than n, which is impossible. The corollary is proved. Proposition 4. Let ft be a gradient-like flow on the closed manifold Mn, n⩾3, and σ be a saddle equilibrium state of the flow ft such that 1) Morse index of the saddle σ equals (n−1); 2) Wuσ∩Wsσ′=∅ for any saddle σ′≠σ. Then clWuσ is a locally flat sphere of dimension (n−1). Proof. By assumption, Wuσ∩Wsσ′=∅ for any saddle equilibrium σ′≠σ, hence, due to Part 3) of Statement 10, there exists a sink equilibrium state ω such that clWuσ=Wuσ∪ω. Therefore, clWuσ is a sphere of dimension (n−1) topologically embedded in Mn. According to Part 2) of Statement 10 the sphere clWuσ is smooth, and, consequently, locally flat at all points of Wuσ.
We prove that the sphere clWuσ is locally flat at the point ω. Let φ:Mn→R be the energy function of the flow ft defined by Statement 9, ε∈(0,1), and Σω be a connected component of the set φ−1(ε) that belongs to Wsω. It follows from Part 4) of Statement 9 that Σω is a smooth sphere of dimension (n−1). Since Wuσ∖σ⊂Wsω, there exists a smoothly embedded (n−1)-ball Bu⊂Wuσ such that σ∈intBu and Σω∩(Wuσ∖Bu)=∅. Then Wuσ∩Σω=Bu∩Σω. Since the function φ strictly decreases along non-singular trajectories of the flow ft, the intersection Bu∩Σω is transversal. Hence the intersection is a smooth closed manifold of dimension n−1+n−1−n=n−2. Since Σω is the section for trajectories of the flow ft, the submanifold Xuσ is the section for all trajectories of the restriction of the flow ft on Wuσ∖σ. Define a retraction by putting into the correspondence to a point x∈Wuσ∖σ the point y=ftx(x)∈Xuσ of the intersection of the trajectory of the flow ft with the set Xuσ passing through the point x. The retraction r can be connected with the identity map id:Wuσ∖σ→Wuσ∖σ by a homotopy therefore, Xuσ is homotopically equivalent to Wuσ∖σ. Since Wuσ∖σ is homeomorphic to Rn−1∖O, it is homotopy equivalent to the sphere of dimension (n−2). Then, due to Poincare Conjecture which was proved for all dimensions, the manifold Xuσ is homeomorphic to the (n−2)-sphere.It follows from the generalized Schoenflies Theorem (see [19], [20]) that there exists a homeomorphism ψu:Σω→Sn−1 such that ψu(Xuσ)=Sn−2=Sn−1∩Ox1…xn−1. Denote by Vω an n-ball laying in Wsω and bounded by Σω. We construct a homeomorphism Ψu:Vω→Bn conjugated the flow ft|Vω with the linear flow at:Rn→Rn, a(x1,…,xn)=((1/2)tx1,…,(1/2)txn). To do this, we assign to each point x∈Vω∖ω a time tx⩽0 such that ftx(x)⊂Σω and set By construction, Ψu(clWuσ∩Vω)⊂Ox1…xn−1. Then the sphere clWuσ is locally flat at the point ω. The proposition is proved. Replacing the flow ft by f−t in Proposition 4, one can immediately obtain the following statement. Corollary 3. Let ft be a gradient-flow on a closed manifold Mn, n⩾3, and σ be a saddle equilibrium state of the flow ft such that 1) Morse index of σ equals 1; 2) Wsσ∩Wuσ′=∅ for any saddle σ′≠σ. Then clWsσ is a locally flat sphere of dimension (n−1). The following statement defines a canonical neighborhood Vp of a saddle hyperbolic equilibrium p, enriched by a coordinate system, in which the flow ft can be represent as a product. The proof of this statement can be found in [21], Ch. 2, § 7, Lemmas 7.2, 7.3. Statement 11. Let p be a hyperbolic equilibrium of the flow ft:Mn→Mn and i∈{1,…,n−1} be the Morse index of p. Then there exists a compact neighborhood Vp⊂Mn of the point p enriched with two continuous maps πs:Vp→Bsp,πu:Vp→Bup, where Bsp=Vp∩Wsp, Bup=Vp∩Wup are balls of dimensions (n−i), i, respectively, containing the point p, with the following properties (Fig. 2): 1) π−1s(p)=Bup, π−1u(p)=Bsp; 2) for any point x∈Bsp (y∈Bup) its preimage π−1s(x) (π−1u(y)) is a smoothly embedded in Mn ball of dimension i (n−i); 3) for any points x∈Bsp, y∈Bup balls π−1s(x), π−1u(u) intersect each other at a single point; 4) ft(π−1s(x))⊃π−1s(ft(x)), f−t(π−1u(y)⊃π−1u(f−t(y)) for any t⩾0. Remark 1. If ft is a gradient-like flow then it is possible to choose the neighborhoods Vp of its saddle equilibria in such a way that Vp∩Vq=∅ for any p≠q. We assume below that the canonical neighborhoods have this property. As a corollary of Statement 11 we obtain a classical fact that any flow in a neighborhood of a hyperbolic equilibria is locally topologically conjugated with a linear flow. Note that this statement is a reformulation of the classical Grobman–Hartman theorem (see [22], [23], and also [21], Ch. 2, § 7, Proposition 2.15). We provide a version of its proof, focusing on the technical fact that the conjugating homeomorphism can be defined on the compact canonical neighborhood Vp. Let bti be a linear flow on the space Rn=Rn−i×Ri given by The flow bti has a unique equilibrium state at the origin O, this equilibrium state is hyperbolic and its Morse index equals i. Set Bn−i={x∈Rn−i:|x|⩽1}, Bi={y∈Ri:|y|⩽1}. Proposition 5. Let p be an equilibrium of a flow ft:Mn→Mn and Morse index of p equals i∈{1,…,n}. Then there exists a homeomorphism hp:Vp→Bn−i×Bi such that hpft=btihp for any t∈R for which the right and left sides are defined. Proof. Let Bsp⊂Wsp, Bup⊂Wup be the balls satisfying the conclusion of Statement 11. From the definition of these balls, it follows that their boundaries ∂Bsp, ∂Bup are secant spheres for the restriction of the flow ft on the sets Wsp∖p, Wup∖p, respectively. Let gs:∂Bsp→∂Bn−i, gu:∂Bup→∂Bi be orientation preserving homeomorphisms. We define homeomorphisms hs:Bsp→Bn−i, hu:Bup→Bi mapping trajectories of the restriction of the flow ft on the balls Bsp, Bup into the trajectories of the restriction of the flow bt on the balls Bn−i, Bi in the following way.
To each point x∈Bsp∖p we assign a time tx⩽0 such that ftx(x)⊂∂Bsp, and set Similarly, we define a homeomorphism hu:Bup→Bi. Namely, to each point x∈Bup∖p we assign a time tx⩾0 such that ftx(x)⊂∂Bup, and set Each point of the set Bn−i×Bi is defined by a pair of coordinates ξ∈Bn−i, η∈Bi. We define the desired homeomorphism hp:Vp→Bn−i×Bi by assigning to each point q∈Vp a point with coordinates (hs(πs(q)),hu(πu(q))). The proposition is proved. § 4. The structure of the non-wandering set and the embedding of invariant manifolds of the flows from the class G(M4)The following proposition proves the first statement of Lemma 1. Proposition 6. Let ft∈G(M4) and the set Ω1ft be non-empty. Then for an arbitrary point p∈Ω1ft the closure clWsp of its stable manifold is a locally flat 3-dimensional sphere that divides the manifold M4 into two connected components. Proof. Let p∈Ω1ft. By assumption, the invariant manifolds of different saddles do not intersect each other. Then, due to Corollary 3, the closure clWsp of the stable manifold Wsp of p is a locally flat 3-dimensional sphere.
We show that the sphere clWsp divides the manifold M4 into two connected components. According to Statement 1, the manifold M4 is orientable. It follows from [20], Theorem 3, that a locally flat sphere Sn−1 in an orientable manifold Mn (n⩾3) is cylindrically embedded, so there exists a closed neighborhood V⊂Mn of the sphere Sn−1 and a homeomorphism h:Sn−1×[−1,1]→V such that h(Sn−1×{0})=Sn−1. By definition, the neighborhood V of the sphere clWsp is divided by this sphere into two parts. We choose points x, y that belong to different connected components of the set V∖clWsp and join them with a smooth arc l⊂V intersecting the sphere clWsp at a single point. If clWsp does not divide M4, then there exists an arc b⊂M4∖Vp connecting the points x, y. By construction, the intersection index of the arc λ=l∪b and the sphere clWsp is equal to 1 or −1 (depending on the choice of the orientation of the arc λ). On the other hand, since the group π3(M4) is trivial by Proposition 1, it is easy to choose the sphere S3⊂M4∖λ homotopic to the sphere clWsp. As the intersection index is a homotopy invariant (see [24], Ch. 2, § 4), the intersection index of the sphere S3 and the arc λ must be equal to ±1, but since S3∩λ=∅, it is zero. This contradiction proves that the sphere clWsp divides the manifold M4 into two connected components. The proposition is proved. Remark 2. Statement 1) of Lemma 1 for a point p∈Ω3ft can be obtained from Proposition 6 by considering the flow f−t. Before we prove statement 2) of Lemma 1, we provide a range of auxiliary statements that also have independent meaning for further constructions. Recall that a set A is called an attractor of a flow ft if there exists a closed neighborhood (a trapping neighborhood) V⊂Mn such that all trajectories of the flow ft intersect its boundary ∂V transversally, and A=⋂t>0ft(V). A set R is called a repeller of the flow ft if it is an attractor for the flow f−t. Let ft∈G(M4). Set Proposition 7. Let ft∈G(M4). Then the set Aft is a connected attractor of the flow ft with the trapping neighborhood Va such that: 1) Va is diffeomorphic to the ball; 2) any trajectory of the restriction of the flow ft on the set Mft intersects the boundary Σ of Va transversely and at a single point. Proof. Let φ:M4→[0,4] be the energy function of the flow ft, defined in Statement 9. Set
Since φ(x)∈[0,1] for any point x∈Aft, Aft⊂intVa. According to Part 2) of Statement 9, trajectories of the flow ft intersect transversely the boundary of the set Va at no more than one point, and ft(Va)⊂intVa for any t>0. Since the set Aft is invariant, then an inclusion Aft⊂⋂t>0ft(Va) holds. Let us show that Aft=⋂t>0ft(Va), whence it will follow that Aft is an attractor, Va is its trapping neighborhood, Mft=⋃t∈Rft(Σ), and any trajectory of the restriction of the flow ft on the set Mft has a non-empty intersection with Σ. Suppose the contrary: let Aft≠⋂t>0ft(Va). Then there exists a point x∈⋂t>0ft(Va)∖Aft. It follows from Statement 10 that there exists an equilibrium state p∈Ωft such that x∈Wup. Since Va⊂f−t(intVa) for any t>0, ⋂t>0ft(Va)=⋂t∈Rft(Va). Therefore the set ⋂t>0ft(Va) is invariant for the flow ft and the whole orbit of the point x belongs to the set ⋂t>0ft(Va). Since the set ⋂t>0ft(Va) is closed, p∈⋂t>0ft(Va). However, all equilibrium states belonging to the set ⋂t>0ft(Va) are only sinks and saddle equilibria that have Morse index equal to one, and whose unstable manifolds lie in the set Aft. Consequently, p∈Aft and x∈Aft, that contradicts to the assumption. Hence Aft=⋂t>0ft(Va). We now prove the connectedness of Va, from which the connectedness of the set Aft as an intersection of connected compact nested sets follows. Suppose that Va is disconnected, that is, can be represented as a union of disjoint non-empty invariant subsets E1,E2. Then the union ⋃t∈Rft(E1∪E2) is also disconnected. However, according to the previous paragraph, ⋃t∈Rft(E1∪E2)=⋃p∈AftWsp. Due to Statement 10, the manifold M4 can be represented as M4=⋃p∈AftWsp∪Rft. Then M4∖Rft=⋃p∈AftWsp, therefore M4∖Rft is disconnected. On the other hand, since the dimension of the set Rft does not exceed 2, Rft does not divide M4, hence the set M4∖Rft is connected. This contradiction proves that Va is connected. We claim that Aft does not contain subsets homeomorphic to the circle. Assume the opposite: let c⊂Aft be a simple closed curve. It follows from Part 3) of Statement 10 that the set Aft∖Ω1ft is a finite set of frames of arcs lying in a disjointed union of the stable manifolds of the sink points. Therefore, the existence of the curve c implies that there exists an equilibrium state p∈Ω1ft belonging to this curve. By statement 1) of Lemma 1, the sphere clWsp divides the ambient manifold M4 into two connected components. Since the one-dimensional unstable separatrices of the point p are separated by the manifold Wsp, they belong to different connected components of the set M4∖clWsp. Therefore, the curve c containing Wup, intersects the sphere clWsp at least at one more point x∈c∩clWsp, different from p. The point x cannot be a source, since Aft, by construction, does not contain sources. The point x cannot be a sink point or a saddle point, since the set Wsp∖p consists of wandering points. Consequently, x belongs to the one-dimensional unstable manifold of some point q∈Ω1ft, that is impossible for Morse–Smale flows. Thus, the set Aft can be represented as a connected graph without cycles, whose vertices are sink points, and whose edges are one-dimensional unstable manifolds of saddle points. Hence, we immediately have |Ω0ft|=|Ω1ft|+1. It follows from Morse theory (see, for example [25]) that the trapping neighborhood Va is obtained from the union of |Ω0ft| pairwise disjoint balls by gluing |Ω1ft| handles of index 1. We prove by induction on the number k=|Ω0ft| that the connected compact manifold that is obtained from k copies of closed n-balls by gluing (k−1) handles of index 1, is diffeomorphic to the n-ball. It will mean that the set Va is diffeomorphic to the n-ball.  Figure 3.The union of two balls with a glued handle: (a) – the manifold is disconnected, (b) – the manifold is connected and diffeomorphic to the ball (to the proof of Proposition 7) For k=1 there is one ball and 0 handles, the statement is true. Assume that the statement is proved for i⩾1 and consider the case k=i+1. Then V is the union of two balls with a glued handle. There are two possibilities to glue a handle, shown in Fig. 3: (a) the manifold is disconnected, which is impossible by assumption; (b) the manifold is connected and diffeomorphic to the ball. The proposition is proved. Proposition 7 and the definition of the set Rft directly imply the following statement. Applying Proposition 7 to the flow f−t proves the following fact. is a connected attractor for the flow f−t for which there exists a trapping neighborhood Vr that is diffeomorphic to a ball and such that Vr∩Va=∅. The attractor Aft, repeller ˜Rft and their trapping neighborhoods are shown on Fig. 4. Proposition 8. Let Σ be the boundary of the trapping neighborhood Va of the attractor Aft. Then: 1) three-dimensional stable and unstable manifolds of saddle equilibria with Morse indices 1 and 3, respectively, intersect the sphere Σ along smoothly embedded two-dimensional spheres; 2) the intersection Wuσ2∩Σ of the sphere Σ with the unstable manifold of the saddle point of the flow ft that has the Morse index 2, is a knot. Proof. Let σi∈Ωift, i∈{2,3}. It follows from Statement 10 that the closure of the set Wuσi consists of Wuσi and the only sink equilibrium state. Since all the sink equilibria of the flow ft belong to the attractor Aft, the intersection is non-empty. Similar to the arguments in the proof of Proposition 4, where the topology of the set Xuσ was studied, one can prove that the set Xuσi is a smooth closed manifold of dimension i−1, which, moreover, is a deformation retract of the set Wuσi∖σi. It follows that for i=2 the set Xuσ2 is a knot, and for i=3 the set Xuσ2 is a two-dimensional sphere.
Let σ1∈Ω1ft. It follows from the definition of the attractor Aft that σ1⊂Aft, so the intersection Wsσ1∩Σ is non-empty. Similarly to the previous case, we prove that the set Wsσ1∩Σ is a two-dimensional sphere. The proposition is proved. Proof. Two cases are possible: 1) the set Ω1ft∪Ω3ft is empty; 2) at least one of the sets Ω1ft,Ω3ft is non-empty.
Consider case 1). Note that it follows from Part 3) of Statement 1 that Euler characteristic of the projective-like manifold of dimension 4 and higher equals 3. Then for the flow ft, the Poincare–Hopf formula implies that the following equality is true: By Corollary 2, the set Ωft contains at least one source and one sink. Since the sources, sinks, and saddles of the flow ft whose Morse index equals 2 make a positive contribution to the left side of the formula (4.1), the condition Ω1ft∪Ω3ft=∅ immediately gives that the set Ωft consists of exactly three equilibrium states: a source, a sink, and a saddle with Morse index 2. Consider case 2). Due to Proposition 7, the trapping neighborhood Va of the attractor is diffeomorphic to the n-ball. By Corollary 5, the set is an attractor of the flow f−t and has a trapping neighborhood Vr diffeomorphic to the ball, and such that Vr∩Va=∅.Remove the interiors of the balls Vr,Va from the manifold M4 and glue to the resulting manifold with boundary two standard n-balls, on one of which the vector field ˙x=x is given, and the field ˙x=−x, x∈Bn, is given on the second, by means of a diffeomorphism φ:∂(M4∖int(Vr∪Va))→Bn×S0. The diffeomorphism φ can be chosen in such a way4 that on the resulting manifold ˜M4 the flow ˜ft∈G(M4) is well defined, whose non-wandering set consists of a source, a sink, and |Ω2ft| saddles of index 2. The operation of gluing balls is equivalent to taking the connected sum with two spheres, so the manifold ˜M4 is homeomorphic to the original manifold M4. Then for the flow ˜ft we obtain the situation considered in case 1). Therefore |Ω2˜ft|=1, hence |Ω2ft|=1. The proposition is proved. The proof of the following statement applies the arguments used in [9], Proposition 3, to prove the local flatness of the closures of invariant manifolds of saddle of index 2 for a gradient-like flow ft∈G(M4) with three states of equilibrium. Proposition 10. Let ft∈G(M4) and Va, Vr be trapping neighborhoods of the attractor and repeller Aft, Rft. Then the knots are trivial. Proof. Due to Statement 11, there exists a compact neighborhood Vp⊂M4 of p and two continuous maps πs:Vp→Bsp, πu:Vp→Bup, where Bsp=Vp∩Wsp, Bup=Vp∩Wup are two-dimensional disks containing the point p, endowing the neighborhood Vp with the direct product structure Bsp×Bup.
The boundary ∂Vp of the neighborhood Vp is the 3-sphere decomposed in a natural way into two solid tori Πs=∂Bsp×Bup, Πu=∂Bup×Bsp with the common boundary T=∂Bsp×∂Bup. Let x∈∂Bsp. Then the curve μ={x}×∂Bup is the meridian of the solid torus Πs and the longitude of the solid torus Πu. It follows from Proposition 7 that for any point x∈Πu there exists a trajectory lx of the flow ft intersecting ∂Va at a unique point zx. Denote by lx,zx the segment of the trajectory lx between the points x, zx, and set Lu=⋃x∈Πulx,zx. By construction, the set Lu is homeomorphic to the direct product B2×S1×[0,1], and the intersection ˜Πu=Lu∩∂Va is a tubular neighborhood of Cu in ∂Va. Moreover, the correspondence x→zx defines a homeomorphism χu:Πu→˜Πu, sending the core ∂Bu of the solid torus Πu to the knot Cu. Denote by Ls, ˜Πs⊂∂W, χs:Πs→˜Πs the similar objects for the solid torus Πs and the sphere ∂Vr. It follows from Proposition 7 that for any point w∈∂Vr∖int˜Πs there exists a trajectory lw of the flow ft intersecting ∂Va∖int˜Πu at a single point vw. Denote by θ:∂Vr∖int˜Πs→∂Va∖int˜Πu a homeomorphism such that θ(w)=vw. It follows from the construction that so θ|∂˜Πs sends the meridian χs(μ) of the solid torus ˜Πs to a longitude χu(μ) of the solid torus ˜Πu.Thus, the sphere ∂Va can be obtained from the sphere ∂Vr by removing the interior of the solid torus ˜Πs and gluing the solid torus ˜Πu by means of the homeomorphism θ, that is, by a non-trivial surgery along the knot Cu. According to Statement 7, the manifold obtained by nontrivial surgery along a nontrivial knot is not homeomorphic to the sphere. Therefore, the knot Cu is trivial. Since the knots Cu, Cs have homeomorphic complements, due to [14], Theorem 1, the knot Cs is also trivial. The proposition is proved. Applying proposition 10 to the flows ft, f−t and using the arguments similar to those in the proof of Proposition 4, we obtain the following statement. Corollary 6. Let ft∈G(M4), p∈Ω2ft. Then the closures of the invariant manifolds of the saddle p are locally flat two-dimensional spheres. Now the statement 2) of Lemma 1 immediately follows from the Proposition 9 and Corollary 6. § 5. Necessary and sufficient conditions of topological equivalence of the flows from the class G(M4)In this Section, Theorem 1 is proved. It follows directly from the definition of the topological equivalence that if flows ft,gt∈G(M4) are topologically equivalent, then their bi-color graphs Γft, Γgt are isomorphic. Suppose that the graphs of the flows ft,gt∈G(M4) are isomorphic by means of the color-preserving isomorphism η:Γft→Γgt which maps the marked vertex of the graph Γft into the marked vertex of the graph Γgt, and prove that the flows are topologically conjugated. Recall that Lft denotes the set of all three-dimensional spheres {clWsp,p∈Ω1ft} and {clWuq,q∈Ω3ft}, and kft is the number of these spheres. By Lemma 1, each sphere from the set Lft divides the manifold M4 into two connected components, so the set M4∖(⋃p∈Ω1ftclWsp∪⋃q∈Ωn−1ftclWuq) consists of mft=kft+1 connected components D1,…,Dmft. Dft denotes the set of all these components. Recall that V(Γft) and E(Γft) denote the set of vertices and edges of the graph Γft, respectively. By the definition of the graph Γft, there are one-to-one correspondence such that the vertices ξ0ft(Di), ξ0ft(Dj) are incident to the edge ξ1ft(L) if and only if the domains Di,Dj∈Dft have a common boundary component L∈Lft.Since graphs Γft,Γgt are isomorphic, they have the same number of vertices and edges, hence mft=mgt. Moreover, the isomorphism η:Γft→Γgt induces a one-to-one correspondence η∗ between the connected components of the sets Dft∪Lft, Dgt∪Lgt as follows: for any D∈Dft, L∈Lft.The one-to-one correspondence η∗ can be extended to a one-to-one correspondence between the sets Ωft and Ωgt as follows. 1. Let σ∈Ω1ft. Then there is a unique point α⊂Ωnft such that clWsσ=Wsσ∪α. Moreover, there is a unique pair of points σ′∈Ω1gt, α′∈Ωngt such that clWsσ′=Wsσ′∪α′ and η∗(clWsσ)=clWsσ′. Set η(σ)=σ′, η(α)=α′. 2. Let σ∈Ωn−1ft. Then there is a unique point ω⊂Ω0ft such that clWuσ=Wuσ∪ω, and a unique pair of points σ′∈Ωn−1gt, ω′∈Ω0gt such that clWuσ′=Wuσ′∪ω′ and η∗(clWuσ)=clWuσ′. We set η(σ)=σ′, η(ω)=ω′. 3. Let ω∈Ω0ft (α∈Ωnft) be such a point that Wsω (Wuα) does not intersect any three-dimensional separatrix of saddle equilibria of the flow ft. Then ω (α) belongs to the unique domain D∈Dft containing the closures of a one-dimensional separatrices of those saddle equilibria whose three-dimensional invariant manifolds form the boundary of the domain D. Moreover, in the domain η∗(D) there is a unique sink (source) point ω′ (α′) of the flow gt, which also does not intersect with any one three-dimensional separatrix of saddle equilibrium states of the flow gt. We set ω′=η∗(ω) (α′=η∗(α)). Everywhere below, for an arbitrary equilibrium state p∈Ωft we denote by p′ an equilibrium state from the set Ωgt such that p′=η∗(p). Recall that in Proposition 7, the section Σ of the restriction of the flow ft to the set Mft=Mn∖(Aft∪Rft) is defined. Denote by Σ′ the similar section for the flow gt. The construction of the homeomorphism H:M4→M4, which maps the trajectories of the flow ft into the trajectories of the flow gt, will be described in steps, each of which will be formulated as a separate proposition. Proposition 11 (Step 1). Let σi∈Ωift, σ′i∈Ωigt, i∈{1,2,3} be saddle equilibria and Vσi, Vσ′i be their canonical neighborhoods defined in Statement 11. Then there is a homeomorphism such that for all t∈R for which the right and left sides of the equality are defined. Proof. By Proposition 5, there is a homeomorphism hσi:Vσi→Bn−i×Bi, conjugating the flow ft|Vσi with the linear flow bti(x,y)=(e−tx,ety), x∈Rn−i, y∈Ri.
We define the desired homeomorphism by hσi,σ′i=h−1σ′ihσi. The proposition is proved. Recall that, according to Statement 11 and Proposition 5, the canonical neighborhood Vσi of any point σi∈Ωift can be represented as a direct product of two balls Bsσi⊂Wsσi, Buσi⊂Wuσi of dimensions (n−i), i, respectively. The boundary of the neighborhood Vσi is represented as the union of two sets with a common boundary as follows: In the case i=1 (i=3) the set ∂Bsσi×Buσi (Bsσi×∂Buσi) is homeomorphic to the annulus S2×[0,1], and the set Bsσi×∂Buσi (∂Bsσi×Buσi) is the union of two 3-dimensional balls. In the case i=2 both sets ∂Bsσi×Buσi, Bsσi×∂Buσi are solid tori. Proposition 12 (Step 2). The spheres Σ, Σ′ can be modified in such a way that the resulting locally flat spheres (which we denote by the same symbols) have the following properties: 1) the spheres Σ, Σ′ are global sections (in the topological sense) for the flows ft|Mft, gt|Mgt, i. e. any trajectory lx∈Mft (lx′∈Mgt) intersects Σ (Σ′) at a single point; 2) for any saddle points σi∈Ωift, σ′i∈Ωigt, the intersections Vσi∩Σ, Vσ′i∩Σ′ are non-empty and consist of sets Proof. Let i=1. Set Π=∂Bsσ1×Buσ1. Suppose that Vσ1∩Σ≠Π. Join the set Π to the sphere Σ by segments of the trajectories of the flow ft and denote by ˜Π the locus of the ends of these trajectories, so that ˜Π⊂Σ. By definition, ˜Π is a submanifold of Σ, so there is an embedding e:∂˜Π×[0,1]→Σ such that e(∂˜Π×[0,1])∩˜Π=e(∂˜Π×{0}) and e(∂˜Π×[0,1])∩Vσ=∅ for any saddle equilibrium different from σ1. On the set ˜Π, a continuous function T:˜Π→R is defined which assigns a time tx such that ftx(x)⊂Π to each point x∈˜Π. We extend this function continuously to the set K=e(∂˜Π×[0,1]) so that T(x)=0 for all points x∈e(∂˜Π×{1}). As a new section, take the following sphere:
We do a similar procedure for all the remaining saddle points, as a result we get a section with the required properties, which we again denote by Σ. We modify the sphere Σ′ in the same way. The proposition is proved. For the equilibrium state σ1∈Ω1ft we set Ssσ1=Wsσ1∩Σ, and for the equilibrium state σi∈Ωift, i∈{2,3}, we set Suσi=Wuσi∩Σ. We use similar notation for saddle equilibria σ′i, i∈{1,2,3}, of the flow gt. Proposition 13 (Step 3). There is a homeomorphism Φ:Σ→Σ′ such that 1) Φ(Ssσ1)=Ssσ′1 for any pair of saddle points σ1∈Ω1ft, σ′=η∗(σ)∈Ω1gt; 2) Φ(Suσi)=Suσ′i for any pair of saddle points σi∈Ωift, σ′i=η∗(σi)∈Ωigt, i∈{2,3}. Proof. Let σ2∈Ω2ft, σ′2∈Ω2gt. Then σ′2=η∗(σ2). By Propositions 8, 10, the sets Suσ2, Suσ′2 are trivial knots. It follows from the definition of a trivial knot that there exists a homeomorphism such that In what follows, we denote the images of the sets Ssσ1, Suσ3 (for all σ1∈Ω1ft, σ3∈Ω3ft) with respect to the homeomorphism Ψ with the same symbols as the originals.
Let D′⊂Σ′ be a 3-ball disjoint from ⋃σ∈Ω1ft∪Ω3ftVσ and such that Suσ′2⊂intD′. Set D=Σ′∖intD′. Proposition 8 implies that for any point σ1∈Ω1ft (σ3∈Ω3ft) the set Ssσ1 (Suσ3) is a smoothly embedded 2-sphere. It follows from the Generalized Schoenflies Theorem that the sphere Ssσ1 (Suσ3) divides the 3-sphere Σ′ into two connected components, the closure of each is a 3-ball. Denote by Dσi, i∈{1,3}, the ball that belongs to the interior of the ball D. We enumerate the saddle points in such a way that for some n0⩽mft the equality holds.We use similar notation for the saddle points of the flow gt. Since the bi-color graphs of the flows ft, gt are isomorphic, the numbering on the set of saddle points of the flow gt can be chosen in a similar way, with σ′i=η∗(σi) for any i∈{1,…,mft}. It follows from Statement 8 that there exists a homeomorphism such that1) Φ0|D′=id; 2) Φ0(Dσi)=Dσ′i, i∈{1,…,n0}. If n0=mft, then we set Φ=Φ0Ψ and go to the next step. Suppose that n0<mft. We denote the images of the balls Dσi, Dσ′i, i∈{1,…,mft}, and their boundaries with respect to the homeomorphism Φ0 by the same characters as the originals. For each ball Dσj, j∈{1,…,n0}, having non-empty intersection with ⋃mfi=n0+1Dσi, we denote by Dσj,1,…,Dσj,nj the pair-wise disjoint discs that are elements of the set Dσj∩⋃mfi=n0+1Dσi and such that ⋃njk=1Dσj,k=Dσj∩⋃mfi=n1+1Dσi, put Dσ′j,k=η∗(Dσ′j,k), k∈{1,…,nj}, and construct a homeomorphism Φj:Σ′→Σ′ identical outside the disk Dσj and such that Φj(Dσj,k)=Dσ′j,k. If Dσj∩⋃mfi=n0+1Dσi=⋃kjl=1Dσj,l, then the required homeomorphism Φ is a superposition of the homeomorphisms Ψ, Φ0 and the constructed homeomorphisms Φ1,…,Φn0. Otherwise, we continue the process and in a finite number of steps we obtain Φ as a superposition of all the constructed homeomorphisms. The proposition is proved. Proposition 14 (Step 4). There exists a homeomorphism HΣ,Σ′:Σ→Σ′ such that HΣ,Σ′|Vσ=hσ,σ′|Vσ for any saddle equilibria σ∈Ωft,σ′∈Ωgt. Proof. Let Φ:Σ→Σ′ be the homeomorphism constructed at Step 3. Set Πσ=Φ(Vσ∩Σ) for any saddle equilibrium σ∈Ωft, Πσ′=Vσ′∩Σ′ for a saddle σ′∈Ωgt, and define a homeomorphism by
Let σ1∈Ω1ft, σ′1=η∗(σ1)∈Ω1gt. The sets Πσ1, Πσ′1 are homeomorphic to the pair of balls B3×S0. Let Π0σ′1⊂Σ′ be a pair of 3-ball such that 1) Πσ1,Πσ′1⊂intΠ0σ′1; 2) Π0σ′1∩Πσ=∅, Π0σ′1∩Πσ′=∅ for any saddle equilibria σ, σ′ of the flows ft, gt, respectively, different from σ1, σ′1. According to Statement 8 there exists a homeomorphism with the following properties:We construct a similar homeomorphism for each saddle point of index 1 and denote by Ψ1 the superposition of the constructed homeomorphisms. Let σ2∈Ω2ft, σ′2=η∗(σ2), Suσ′2=Wuσ′2∩Σ′. In this case the sets Πσ2, Πσ′2 are solid tori. Let Π0σ′2⊂Σ′ be a tubular neighborhood of the knot Suσ′2 such that Πσ2,Π′σ′2⊂intΠ0σ′2, Π0σ′2∩Πσ=∅, Π0σ′2∩Πσ′=∅ for any saddles σ, σ′ different from σ2, σ′2. By definition, the homeomorphism h′|σ,σ′ maps the longitude and the meridian of the solid torus Πσ2 into the longitude and the meridian of the solid torus Π′σ′2, correspondingly, hence, by Proposition 2, there exists a homeomorphism such thatLet σ3∈Ω3ft, σ′3=η∗(σ)∈Ω3gt. The sets Πσ3, Πσ′3 are homeomorphic to the direct product S2×[0,1]. Let Π0σ′3 be a tubular neighborhood of the set Ssσ′3=Wuσ′3∩Σ′ with properties similar to ones of the sets Π0σ′1, Π0σ′2. According to Proposition 3, there exists a homeomorphism such thatWe construct a similar homeomorphism for each saddle point of index 3 and denote by Ψ3 the superposition of the constructed homeomorphisms. Now the desired homeomorphism HΣ,Σ′ is defined as a superposition Ψ3Ψ2Ψ1Φ. The proposition is proved. Proposition 15 (Step 5). The homeomorphism HΣ,Σ′:Σ→Σ′ can be extended to a homeomorphism H:M4→M4 such that Hft=gtH. Proof. For any saddle point σ(σ′) of the flow ft(gt) set and define a homeomorphism by where tx∈R is the time such that ftx(x)∈Vσ.
For any point x∈Mft set where tx∈R satisfies ftx(x)∈Σ.By construction, the homeomorphism HMft,Mgt coincides with the homeomorphism Hσ,σ′ on the intersection Mft∩Vσ, so the following formula defines a homeomorphism which can be unequivocally extended to the set Ω0ft∪Ω4ft: The proposition is proved.§ 6. Realization of the topological equivalence classes of the flows from the class G(M4) Proof. Each edge e of the graph Γft corresponds to an (n−1)-dimensional sphere which, by virtue of Lemma 1, divides the supporting manifold M4 into two connected components. Therefore, the edge e divides the graph Γft into two connected components, hence the graph Γft does not contain cycles.
We now show that the graph Γft is connected. By definition, the graph Γft has kft edges and kft+1 vertices. It is known that a connected graph with k+1 vertices is a tree if and only if it has exactly k edges. Assume that the graph Γf is not connected. Then it consists of connected subgraphs Γ1,…,Γm, m⩾2, and adding m−1 edges turns the set of subgraphs Γi into a connected graph without cycles (i. e. the tree) with kf+1 vertices and kf+m edges, that contradicts the mentioned property of trees. Therefore, the graph Γft is connected and does not contain cycles, that is, it is a tree. The proposition is proved. Proof of Theorem 2. Let Γ be an arbitrary tree with one marked vertex, whose edges are colored in two colors s and u. Without loss of generality, suppose that the marked vertex is incident to at least one edge of color s. In the opposite case, we reverse the colors of the edges (replacing s with u and vice versa) and construct the flow ft with the resulting graph. The desired flow will be the flow f−t.
Denote by gt1∈G(M4) the flow whose set of saddle equilibria consists of exactly one saddle (then its Morse index equals 2). The algorithm of constructing such a flow is described in [9], [10]. Let ω∈Ω0gt1, σ∗∈Ω2gt1. Denote by gt2 a gradient-like flow on the sphere S4 whose bi-color graph Γgt2 is isomorphic to the graph Γ. The algorithm of constructing such flows is described in paper [27]. Let D∈S4 be the domain corresponding to the marked vertex of the graph Γ. By assumption, the boundary of the domain includes at least one stable manifold of dimension n−1 of the saddle point σ∈Ω1gt2. Consequently, the boundary of D also includes the source equilibrium state α, which belongs to the closure of the manifold Wsσ. Denote by Sn−1ω⊂Wsω, Sn−1α⊂Wuα secant spheres to the restriction of the flows gt1, gt2 to the sets Wsω∖ω , Wuα∖α, respectively (such spheres can be chosen as level hypersurfaces of the Morse energy functions for the flows gt1, gt2) and by Bnω,Bnα the balls bounded by the spheres Sn−1ω, Sn−1α such that ω∈Bnω, α∈Bnα. Choose the orientation of the spheres Sn−1ω, Sn−1α as of the boundaries of the balls Bnω, Bnα. Denote by φ:Sn−1ω→Sn−1α a diffeomorphism reversing the chosen orientation and such that φ(Sn−1ω∩Wsσ∗)⊂D∩Sn−1α, glue the manifolds M4∖intBnω, S4∖intBnα by means φ, denote by ˜M4 the obtained manifold and by p:(M4∖intBnω)∪S4∖intBnα the canonical projection. The manifold ˜M4 is the connected sum of the complex projective plane M4 and the sphere S4, so it is homeomorphic to M4. A slight modification of the flows gt1, gt2 in the neighborhood of the spheres Sn−1ω, Sn−1α (see details in [26]) allows us to define a flow ft∈G(˜M4) on the manifold ˜M4 such that ft coincides with the flow gt1 on the set p(M4∖intBnω) and with the flow gt2 on the set S4∖intBnα. By construction, the graph Γft is isomorphic to the graph Γ. Theorem 2 is proved. 
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\Bibitem{GriGur22}
This publication is cited in the following 2 articles:
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