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This article is cited in 3 scientific papers (total in 3 papers) A combinatorial invariant of gradient-like flows on a connected sum of V. Z. Grines, E. Ya. Gurevich National Research University Higher School of Economics, Nizhnii Novgorod, Russia
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     § 1. Introduction and statement of the resultsWe recall that a flow ft on a smooth closed manifold Mn is called gradient-like if its nonwandering set consists of a finite number of hyperbolic saddle equilibria, and the invariant manifolds of saddle equilibria intersect each other transversally. Smale [1] showed that the gradient flow of a Morse function (a smooth function every critical point of which is nondegenerate) is gradient-like for some choice of the metric. A Morse function exists on any manifold, and so each manifold admits a gradient-like flow. It was shown in [2] and [3] that any gradient-like flow satisfies Morse’s inequalities relating the structure of the nonwandering set of the flow with the topology of the ambient manifold. In particular, the following relation holds. Let ci be the number of equilibria such that the dimension of their unstable manifolds (Morse index) is i∈{0,…,n}, and let χ(Mn) be the Euler characteristic of the manifold Mn. Then Note that by (1.1) the number |χ(Mn)| is a lower bound for the number of saddle equilibria. If n=2 and M2 is orientable, then, as is known, M2 is homeomorphic to the connected sum of the 2-sphere S2 and g⩾0 tori T2=S1×S1; it is also known that, χ(M2)=2−2g. In this case it follows from (1.1) that the genus g of the manifold M2 can be expressed in terms of the number of nodal equilibria νft=c0+c2 and the number of saddle equilibria μft=c1 as follows:For n>2 the Euler characteristics is no longer a complete topological invariant; moreover, if n is odd, then χ(Mn)=0 for any manifold Mn, and therefore, in general, formula (1.1) provides very little information, but we have at out disposal Assertion 1 (see below), which strengthens this formula considerably. Let G(Mn) be the class of gradient-like flows defined on a closed connected orientable manifold Mn of dimension n⩾3 such that, for any ft∈G(Mn), the invariant manifolds of different saddle equilibria are disjoint. We denote by Sng the manifold homeomorphic to the connected sum of the sphere Sn and g copies of the direct product Sn−1×S1.Let ft∈G(Mn), n⩾3, and let νft, μft be, respectively, the number of nodal and saddle equilibria of the flow ft. We also set gft=(μft−νft+2)/2. The following result holds. Assertion 1. 1. If the Morse index of each saddle equilibrium of a flow ft is 1 or n−1, then gft is a positive integer, and the ambient manifold Mn is Sngft. 2. If the ambient manifold Mn is Sng, then the Morse index of each saddle equilibrium of ft is equal to 1 or n−1 and, in addition, g=gft. The first claim in Assertion 1 follows from [4] and [5], where analogous facts for Morse-Smale cascades were proved. Claim 2 in the case g=0 was proved in [6] and, for g>0, in [7]. According to [6], the phase diagram is a complete topological invariant for G(Sn)-flows, for n>3. Recall that the phase diagram is a combinatorial invariant, which extends the Leontovich–Maier scheme of a dynamical system and the Peixoto graph, which were applied in Ch. 11 of [8] and [9] to the topological classification of two-dimensional Morse-Smale cascades. The phase diagram of a flow ft is a directed graph whose vertex set is isomorphic to the set of saddle equilibria of the flow, and whose edge set is isomorphic to the set of separatrices of saddle equilibria; in addition, an edge connects vertices p and q directly from p to q (from q to p) if and only if p corresponds to a saddle equilibrium and q corresponds to a sink or a source (Figure 1). This paper is concerned with the solution of the problem of topological classification of G(Sng)-flows for g>0 and n⩾3. For n=3 a topological classification of G(S3g)-flows follows from more general results due to Umanskii (see [10]), who also used an invariant similar to the Leontovich-Maĭer scheme of a dynamical system. Figure 2 shows that the phase diagram is not a complete topological invariant of G(Sng)-flows in the case g>0 (for any n⩾3). In Figure 2 we show the phase portraits and phase diagrams of two G(Sn1)-flows. The Morse index of the saddle equilibria σ1, σ3 and σ4 (σ′1, σ′3 and σ′4) is 1, the Morse index of the saddle σ2 (respectively, σ′2) is n−1. The union clWsσ1∪clWuσ2 (clWsσ′1∪clWuσ′2) of the stable and unstable manifolds of σ1 and σ2 (σ′1 and σ′2), respectively, divides the ambient manifold Sn1=Sn−1×S1 into two connected components D1 and D2 (respectively, D′1 and D′2). The equilibria σ3 and σ4 of the flow ft lie in the same connected component D1, and the saddle equilibria σ′3 and σ′4 of f′t lie in different connected components D′1 and D′2, respectively. Hence there exists no homeomorphism of Sn1 onto itself that maps trajectories of ft to those of f′t. We show below that a classification of G(Sng)-flows for g>0 in combinatorial terms is possible, provided that as an invariant, one considers a bi-colour graph similar to that introduced by Oshemkov and Sharko [11] for the classification of Morse-Smale flows on surfaces. Note that a bi-colour graph was also used in [12] for the classification of Morse-Smale cascades on the sphere Sn of dimension n⩾4. Let Ωift be the set of saddle equilibria of the flow ft whose Morse index is i∈{0,1,n−1,n}. Let ft∈G(Sng) and σ1∈Ω1ft (σn−1∈Ωn−1ft). It follows from Theorem 2.3 in [13] that the closure of a stable (respectively, unstable) manifold of a saddle equilibrium of the flow ft contains, in addition to the manifold itself, a unique point, which is a source (sink) equilibrium. Hence the set Lsσ1=clWsσ1 (Luσn−1=clWuσn−1) is a sphere of dimension n−1. We denote the set of all spheres {Lsσ1,Luσn−1,σ1∈Ω1ft,σn−1∈Ωn−1ft} by Lft and the set of all connected components of the manifold Sng∖(⋃σ1∈Ω1ftLsσ1∪⋃σn−1∈Ωn−1ftLuσn−1) by Dft. Definition 1. The bi-colour graph of a flow ft∈G(Sng) is the graph Γft with the following properties:
Definition 2. Graphs Γft,Γf′t of flows ft,f′t∈G(Sng) are isomorphic if there exists a bijection ζ:V(Γft)→V(Γf′t) preserving adjacency and the colours of edges. A comparison of Figures 2 and 3 shows that bi-colour graphs (unlike phase diagrams) are capable of distinguishing the mutual arrangement of the closures of the separatrices of topologically nonequivalent flows ft and f′t. Theorem 1. Two flows ft∈G(Sng) and f′t∈G(Sng′) are topologically equivalent if and only if their graphs Γft and Γf′t are isomorphic. § 2. Auxiliary results2.1. Embeddings of spheres in a manifold and extensions of local homeomorphismsWe set By a ball or a disc of dimension n⩾1 we mean a manifold Bn homeomorphic to Bn. A sphere of dimension n−1 (an open ball of dimension n) is a manifold homeomorphic to the boundary Sn−1 (the interior intBn) of the standard ball Bn. A continuous map f:X→Mn is a topological embedding if f:X→f(X) is a homeomorphism (here f(X) is equipped with the topology induced from Mn). The image f(X) is called a topologically embedded manifold. Let f:Sn−1→Mn be a topological embedding. The sphere Sn−1=f(Sn−1) is called locally flat if, for any point x∈Sn−1, there exist a neighbourhood Ux⊂Mn and a homeomorphism ψ:Ux→Rn such that ψ(Sn−1∩Ux) is a linear subspace of Rn of dimension n−1. The following classical Brouwer theorem [14] generalizes the well-known Jordan theorem (which claims that any simple closed curve on the plane divides this plane into two connected components; for a proof, see, for example, [15]). Assertion 2 (Jordan-Brouwer theorem). Let φ:Sn−1→Sn be a topological embedding, Sn−1=φ(Sn−1), n>0. Then the set Sn∖Sn−1 consists of two connected components. Corollary 1. Let Sn−11,…,Sn−1m⊂Sn, m⩾1, be pairwise disjoint topologically embedded spheres,. Then the set Sn∖⋃mi=1Sn−1i has precisely m+1 connected components. Proof.
Let us prove this result using induction on the number of spheres. For m=1 the result is true by the Jordan-Brouwer theorem. Assuming that the required result holds for all m∈{1,…,i}, let us verify it for m=i+1.
Since m is finite, there exists j∈{1,…,m} such that the sphere Sn−1j divides Sn into two connected components Vj and Wj such that all spheres in the set L=⋃mk=1Sn−1k∖Sn−1j lie in the same component, which we denote by Vj. By the induction assumption the set Sn∖L consists of (m−1)+1=m components X1,…,Xm. The set Wj∪Sn−1j=clWj is connected, and thus it lies fully in one of the connected components of Sn∖L. We denote this component by Xm. The set Xm∖Sn−1j has precisely two connected components Vj∩Xm and Wj. Therefore, Sn∖⋃mk=1Sn−1k has precisely m+1 connected components X1,…,Xm−1, Vj∩Xm and Wj. We recall that Schoenflies’s theorem (see [16] and [17]) asserts that any simple curve on the plane R2 (or the sphere S2) is the boundary of a 2-disc in R2 (in S2). The following result holds (see [18]). Assertion 3 (generalized Schoenflies theorem). If the sphere Sn−1 is locally flatly embedded in Sn, n⩾3, then the closures of the connected components of the complement to Sn−1 are n-balls. Assertion 4 (Annulus Theorem). Let Sn−10 and Sn−11 be disjoint locally flat (n−1)-spheres in Sn, and let Kn be an open domain in Sn bounded by Sn−10 and Sn−11. Then the closure of the domain Kn is homeomorphic to the annulus Sn−1×[0,1]. Brown and Gluck (see Theorem 9.4 in [19]; also see [20], Chap. 5, Theorem 3.2) showed that the Annulus Theorem is closely related to Conjecture SHCn on stable homeomorphisms of the sphere Sn. A homeomorphism f:Sn→Sn is called stable if it can be represented as a finite composition of homeomorphisms each of which is identical on some open set. Conjecture SHCn (now verified; see the references below) claims that each orientation-preserving homeomorphism of the sphere Sn is stable. Brown and Gluck proved that Conjecture SHCn implies the Annulus Theorem in dimension n, and, in its turn, the Annulus Theorem in dimensions k⩽n implies Conjecture SHCn. For n⩽3 Conjecture SHCn was proved in [21]. This implies that the conclusion of the Annulus Theorem holds for dimensions 2 and 3 (however, the two-dimensional Annulus Theorem can easily be obtained from Schoenflies’s theorem; see Ch. 2, § A, in [22]). For n>4 Conjecture SHCn was proved by Kirby in 1969 (see [23]). The Annulus Theorem for n=4 was established by Quinn in 1984 (see [24] and the clarifications of this result in [25]; also see the survey [26]). This implies the validity of Conjecture SHC4. The following result is a consequence of Conjecture SHCn (see § 4 in [19]). Assertion 5. Any orientation-preserving homeomorphism of the sphere Sn, n⩾1, is isotopic to the identity.1 Let M and N be n-dimensional closed orientable manifolds with boundaries, n⩾1, X⊂∂M and Y⊂∂N be closed homeomorphic submanifolds of dimension n−1, and g:X→Y be a homeomorphism reversing the natural orientation of the boundary. Consider an equivalence relation on the union M∪N: if x∈M∪N∖(X∪Y), then x∼x; if x∈X and y∈Y, then x∼g(x) and y∼g−1(y). The quotient space by this equivalence relation is a topological manifold. We say that this manifold is obtained by gluing M and N by means of the homeomorphism g:X→Y.The following classical theorem, sometimes called Alexander’s trick after J. W. Alexander, has many applications. Assertion 6 (extension of a homeomorphism from the sphere to the ball). Let Bn1 and Bn2 be balls of dimension n, and let h:∂Bn1→∂Bn2 be an arbitrary homeomorphism. Then there exists a homeomorphism H:Bn1→Bn2 such that H|∂Bn1=h|∂Bn1. Proof. Let h1:Bn1→Bn and h2:Bn2→Bn be arbitrary homeomorphisms. Let the homeomorphism ˜h:∂Bn→∂Bn be defined by ˜h=h2hh−11|∂Bn. We also consider the homeomorphism ˜H:Bn→Bn defined by ˜H(rx)=r˜h(x) for each radius vector x∈∂Bn and r∈[0,1]. Now the required homeomorphism H is given by the formula H=h−12˜Hh1, which completes the proof. Corollary 2. Let Bn1 and Bn2 be two balls of dimension n⩾1, g:∂Bn1→∂Bn2 be a homeomorphism changing the natural orientation of the boundary and Mn be the manifold obtained from the union Bn1∪Bn2 by gluing by means of g. Then Mn is homeomorphic to the sphere Sn. Proof. Let Dn1={(x1,…,xn+1)∈Sn∣xn+1⩾0} and Dn2={(x1,…,xn+1)∈Sn∣xn+1⩽0}, and let h1:Bn1→Dn1 be an arbitrary homeomorphism. By Assertion 6 there exists a homeomorphism h2:Bn2→Dn2 such that h2|∂Bn2=h1g−1|∂Bn2. Consider the continuous map H:Bn1∪Bn2→Sn defined by The map H is a homeomorphism on intBn1∪intBn2 such that H(x)=h1(x)=h2(g(x))=H(g(x)) for all points x∈∂Bn1 and g(x)∈∂Bn2. Therefore, H induces a homeomorphism Bn1∪gBn2→Sn. This proves the corollary. Proposition 1. Let M be a topological manifold with boundary, X be a connected component of the boundary and N be a manifold homeomorphic to X×[0,1] and such that M∩N=∂M∩∂N=X. Then the manifold M∪N is homeomorphic to M. Proof. By Theorem 2 in [28] there exists a topological embedding h0:X×[0,1]→M such that h0(X×{1})=X. We set M0=h0(X×[0,1]). Let h1:X×[0,1]→N be a homeomorphism such that h1(X×{0})=X=h0(X×{1}).
To complete the proof it suffices to consider the homeomorphisms g:X×[0,1]→X×[0,1] and ˜h1:X×[0,1]→N, h:X×[0,1]→M0∪N defined by and respectively, and define the homeomorphism H:M∪N→M byProposition 1 is proved. The next result is a direct consequence of Proposition 1 and Corollary 2. Corollary 3. The connected manifold obtained by attaching the disjoint union of two balls Bn+ and Bn− to the annulus Sn−1×[0,1] is homeomorphic to the sphere Sn. Proposition 2 (extension of a homeomorphism from the boundary to the interior of the annulus). Let Kn=Sn−1×[0,1], and let ψ0:Sn−1×{0}→Sn−1×{0} and ψ1:Sn−1×{1}→Sn−1×{1} be orientation-preserving homeomorphisms. Then there exists a homeomorphism Ψ:Kn→Kn such that 1) Ψ|Sn−1×{0}=ψ0|Sn−1×{0}, Ψ|Sn−1×{1}=ψ1|Sn−1×{1}; 2) Ψ(Sn−1×{1/2})=Sn−1×{1/2}. Proof. By Assertion 5 there exist isotopies H0:Sn−1×[0,1]→Sn−1×[0,1] and H1:Sn−1×[0,1]→Sn−1×[0,1] such that
1) H0|Sn−1×{0}=ψ0, H0|Sn−1×{1}=id; 2) H1|Sn−1×{0}=ψ1, H1|Sn−1×{1}=id. To complete the proof it suffices to define the homeomorphism Ψ:Sn−1×[0,1]→Sn−1×[0,1] by Proposition 3. Let ei,e′i:Bn→intBn, i∈{1,…,k}, be orientation-preserving topological embeddings such that 1) the spheres ei(∂Bn) and e′i(∂Bn) are locally flat in Bn for all i∈{1,…,k}; 2) ei(Bn)∩ej(Bn)=∅ and e′i(Bn)∩e′j(Bn)=∅ for all 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}. Proof. In the case of smooth embeddings ei and e′i, the result of the proposition is secured by Theorems 3.1 and 3.2 in Ch. 8 of [29]. We give a proof independent of smoothing arguments. We argue by induction on k.
Let k=1. It follows from Assertion 4 that the sets K1=Bn∖e1(intBn) and K′1=Bn∖e′1(intBn) are homeomorphic to the standard annulus Sn−1×[0,1]. Let φ1:Sn−1×[0,1]→K1 and φ′1:Sn−1×[0,1]→K′1 be orientation-preserving homeomorphisms such that φ1(Sn−1×{0})=φ′1(Sn−1×{0})=∂Bn. Let the maps ψ1:Sn−1×{0}→Sn−1×{0} and η1:Sn−1×{1}→Sn−1×{1} be defined by respectively. By definition, both ψ1 and η1 preserve orientation, and therefore there exist isotopies ψ1,t:Sn−1→Sn−1 and η1,t:Sn−1→Sn−1, t∈[0,1], joining them with the identity map. Let ε∈(0,1/3). Let the homeomorphisms G1:Sn−1×[0,1]→Sn−1×[0,1] and h1:Bn→Bn be defined by and respectively.It is easily checked that for k=1 the homeomorphism h1 is the required one. Now assume that we have already constructed a homeomorphism hj:Bn→Bn such that: 1) hj|∂Bn=id; 2) hjei=e′i, 1⩽i⩽j. Let us construct a homeomorphism hj+1:Bn→Bn satisfying 1) hj+1|∂Bn=id; 2) hj+1ei=e′i, 1⩽i⩽j+1. We set Kj+1=Bn∖ej+1(intBn) and K′j+1=Bn∖e′j+1(intBn) and denote by φj+1:Sn−1×[0,1]→Kj+1 and φ′j+1:Sn−1×[0,1]→K′j+1 orientation-preserving homeomorphisms such that φj+1(Sn−1×{0})=φ′j+1(Sn−1×{0})=∂Bn. Let the maps ψj+1:Sn−1×{0}, ηj+1:Sn−1×{1} be defined by respectively. By definition ψ and η preserve orientation, and so there exit isotopies ψj+1,t:Sn−1→Sn−1 and ηj+1,t:Sn−1→Sn−1, t∈[0,1], joining them with the identity map. Let ε∈(0,1/3) be such that the set C={(x,t)∣x∈Sn−1, t∈[0,ε]∪[1−ε,1]} is disjoint from the balls φj+1(ei(Bn)) and φ′j+1(e′i(Bn)) for all i∈{1,…,j}. Let the homeomorphisms Gj+1:Sn−1×[0,1]→Sn−1×[0,1] and hj+1:Bn→Bn be defined by and respectively.A direct verification shows that hj+1 is the required homeomorphism. Proposition 3 is proved. Let ei:Bn→intBn, i∈{1,…,k}, be topological embeddings satisfying the assumptions of Proposition 3. A manifold with boundary that is homeomorphic to the manifold Bn∖⋃ki=1ei(intBn) will be called an n-ball with k holes or an n-sphere with k+1 holes. The next result is a direct consequence of Proposition 3. 2.2. HandlebodiesA handle of dimension n and index k is the direct product Hnk=Bk×Bn−k. We say that an n-manifold M is obtained from an n-manifold N with boundary ∂N by attaching a handle Hnk if there exists an embedding ψ: ∂Bk×Bn−k→∂N such that M=N⋃ψHnk. A manifold Qnk obtained from the ball Bn by attaching handles of index ⩽k is called an (n,k)-handlebody. The case k=1 is of special interest. The number g of handles of index 1 of a handlebody Qn1 is the genus of this body. 2.3. Some facts from graph theoryA graph Γ is a union of two sets: a finite set V(Γ) of elements called vertices and a finite set E(Γ) of elements called edges. The edge set of a graph is formed by some pairs of its (not necessarily distinct) vertices. Two vertices v,w∈V(Γ) are adjacent if the pair v, w belongs to the set E(Γ); in this case the edge e=(v,w) is said to be incident to the vertices v and w, and v and w are incident to e. The vertex set of a graph can be visualized as a point set in some Euclidean space (or a subset of it), and the edge set can be regarded as a set of arcs joining adjacent vertices. In view of this analogy an edge (v,w) is said to join the vertices v and w, and v and w are called end vertices of the edge (v,w). A graph Γ is simple if the end vertices of any edge of it are distinct. In what follows we consider only simple graphs without special mention. A simple path or a route connecting vertices v and w is a sequence of pairwise distinct vertices v0=v,v1,…,vk=w, where each successive vertex is edge-connected with the previous one. A simple cycle is a route connecting a vertex v with itself. A graph Γ is called connected if any two vertices in it are connected by a route. A cycle-free graph is called acyclic. A connected acyclic graph is called a tree. The main properties of trees are listed in the following statement (see [30], Theorem 13.1). Assertion 7. Let Γ be a graph with r edges and b vertices. Then the following assertions are equivalent: 1) Γ is a tree; 2) Γ is a connected graph and b=r+1; 3) Γ is an acyclic graph and b=r+1; 4) any two different vertices are connected by a unique simple route; 5) Γ is an acyclic graph such that if an arbitrary pair of its distinct vertices is complemented by an edge joining them, then the resulting graph contains precisely one cycle. The number of edges incident to a vertex is called the degree of this vertex. A vertex of degree 1 is a leaf. For a tree Γ, the rank of a vertex is defined as follows. We associate with Γ a sequence of trees Γ0=Γ,Γ1,…,Γr such that, for any i∈{1,…,r}, the tree Γi is obtained from Γi−1 by removing all leaves and all edges incident to these leaves, and the tree Γr consists either of one vertex or of two vertices joined by an edge. In the first case Γ is called a central tree, and in the second case, a bicentral tree. The vertices and edge (if exists) of the tree Γr are called the central vertices and edge of Γ. The rank of a vertex v∈V(Γ) is defined by This definition shows that if an edge (v,w) is not central, then |rank(v)−rank(w)|=1, and the central vertices of a bicentral tree have the same rank r. Figure 4 shows central and bicentral trees and indicates the rank of each of their vertices.Two graphs Γ and Γ′ are called isomorphic if there exists a bijection ξ:V(Γ)→V(Γ′) (called an isomorphism) such that if two vertices v,w∈V(Γ) are adjacent, then so are also the vertices ξ(u),ξ(v)∈V(Γ′). § 3. Necessary and sufficient conditions for the topological equivalence of G(Sng)-flows3.1. The scheme of a flow as a topological invariantLet φ:Mn→R be a Cr-smooth function on a manifold Mn, r⩾2. A point p∈Mn is called a critical point of φ if in local coordinates in a neighbourhood of the point p. A point q∈Mn that is not critical is called a regular point of φ. A critical point p of φ is called nondegenerate if the Hessian matrix (∂2φ∂xi∂xj)|p is nonsingular.A function φ:Mn→R is a Morse function if all of its critical points are nondegenerate. The next result, which follows from [1] and [31], defines the self-indexing energy function, which is an efficient tool for examining gradient-like flows. Set Let σ1∈Ω1ft, σn−1∈Ωn−1ft be arbitrary points. According to Theorem 2.3 in [13], there exist points ω∈Ω0ft and α∈Ωnft such that clWsσ1=Wsσ1∪α and clWuσn−1=Wuσn−1∪ω. Since the function φ decreases along the trajectories of the flow ft, the sets Wsσ1∩Σft and Wuσn−1∩Σft are nonempty. We set Definition 3. The triple Sft={Σft,Csft,Cuft} is called the scheme of the flow ft∈G(Sng), and the set Σft is called its characteristic section. In Lemma 1 below we show that the topology of the section Σft and the mutual arrangement of the connected components of the set Csft∪Cuft on Σft determine fully the class of topological equivalence of the flow ft∈G(Sng), and in § 3.3 we show that a bi-colour graph is capable of describing uniquely the mutual arrangement of these connected components. Definition 4. Two schemes Sft={Σft,Csft,Cuft} and Sf′t={Σ′f′t,Csf′t,Cuf′t} are called equivalent if there exists a homeomorphism h:Σft→Σ′f′t such that Lemma 1. Two flows ft,f′t∈G(Sng) are topologically equivalent if and only if so are their schemes Sft and Sf′t. Proof. Set Aft=Ω0ft∪WuΩ1ft, Rft=Ωnft∪WsΩn−1ft and Vft=Sng∖(Aft∪Rft) and denote the analogous objects for the flow f′t by Af′t, Rf′t and Vf′t.
Let us prove the necessity in the lemma. Assume that two flows ft,f′t∈G(Sng) are topologically equivalent via a homeomorphism ˜h. Since Σ′f′t is a level surface of the energy function of the flow, which is strictly decreasing on Vf′t, for each point y∈Vf′t there exists a unique point ty∈R such that f′ty(x)⊂Σ′f′t. We define the homeomorphism h:Σft→Σ′f′t by Let x lie in the manifold Wsσ1 of some saddle point σ1∈Ω1ft. Then ˜h(σ1)∈Ω1f′t, ˜h(Wsσ)=Wsh(σ1) and h(Wsσ1∩Σft)=Ws˜h(σ1)∩Σ′f′t. A similar analysis shows that h(Wuσn−1∩Σft)=Wu˜h(σn−1)∩Σ′f′t for any point σn−1∈Ωn−1ft. So the schemes Sft and Sf′t are equivalent. Let us prove sufficiency in the lemma. Assume that the schemes Sft and Sf′t are equivalent and h:Σft→Σ′f′t is a homeomorphism such that h(Csft)=Csf′t and h(Cuft)=Cuf′t. For any point x∈Σft we set x′=h(x) and denote by Ox and O′x′ the trajectories of the flows ft and f′t passing through the points x and x′, respectively. Let φ,φ′:Sng→[0,n] be the self-indexing energy functions of ft and f′t, respectively. For any point y∈Vft there exist a unique point x∈Σft and a unique number cy∈(0,n) such that y=Ox∩φ−1(cy). Let the homeomorphism H:Vft→Vf′t be defined by The homeomorphism h maps the set Csft (Cuft) to the set Csf′t (to Cuf′t, respectively) and so the homeomorphism H thus constructed sends all stable ( unstable) separatrices of dimension n−1 of saddle equilibria of the flow ft to stable (unstable) separatrices of dimension n−1 of saddle equilibria of f′t. Hence the homeomorphism H extends uniquely to the set of all saddle equilibria. Now, keeping the same notation H for the resulting homeomorphism, we extend it to the one-dimensional separatrices of saddle equilibria. To this end we note that, for any c∈(0,1), and ⋃σ∈Ω1ftWuσ∩φ−1(c) and ⋃σ′∈Ω1f′tWuσ′∩φ′−1(c) are sets of the same cardinality. Hence H extends to ⋃σ∈Ω1ftWuσ by continuity. In a similar way H extends to ⋃σ∈Ωn−1ftWsσ and further to Ω0ft∪Ωnft.Lemma 1 is proved. 3.2. The topology of the characteristic section Σft and of connected components of Csft∪CuftWe recall that, given a flow ft∈G(Sng), we denote by νft and μft the numbers of nodal and saddle equilibria, respectively; the quantity gft is defined by By Assertion 1 we have g=gft. Lemma 2. Let ft∈G(Sng). Then the following hold. 1. The characteristic section Σft is the boundary of an (n,1)-handlebody of genus g. 2. For all σ1⊂Ω1ft and σn−1⊂Ωn−1ft each connected component lsσ1=Wsσ1∩Σft and luσn−1=Wuσn−1∩Σft of the set Csft or Cuft, respectively, is a sphere of dimension n−2 which is smoothly embedded in Σft. 3. If g>0, then there exist two sets of saddle equilibrium states σ11,…,σg1∈Ω1ft and σ1n−1,…,σgn−1∈Ωn−1ft and sets {Tsi⊂Σft} and {Tuj⊂Σft} of pairwise disjoint tubular neighbourhoods of the spheres {lsσi1} and {luσjn−1}, respectively, such that the sets cl(Σft∖⋃gi=1Tsi) and cl(Σft∖⋃gj=1Tuj) are homeomorphic to the sphere with 2g holes. Proof. Let us prove part 1. We set Qa=φ−1([0,n/2]), where φ is the self-indexing energy function of the flow ft. By definition Σft=∂Qa. First we prove that Qa is connected.
Since φ decreases along the trajectories of ft, we have Qa⊂⋃p∈Ω0ft∪Ω1ftWsp. We set Ua=⋃i∈Zfi(Qa) and Rft=Ωnft∪WsΩn−1ft. According to Theorem 2.3 in [13], the manifold Sng can be represented as Sng=⋃p∈Ω0ft∪Ω1ftWsp∪Rft. Hence Ua=Sng∖Rft. Since n⩾3 and the dimension of the set Rft is at most 1, it follows that Ua is connected (see [32], Ch. 4, Theorem 4). Hence Qa is too. Indeed, if Qa were disconnected, then it could be represented as the union of nonempty disjoint closed subsets E1 and E2 such that if x∈Ei, then ft(x)∈Ei for any t⩾0. Hence the set Ua=⋃t∈R(ft(E1)∪ft(E2)) would also be disconnected (if the intersection ⋃t∈Rft(E1)∩⋃t∈Rft(E2) is nonempty, then for any point x in this intersection there exists tx such that ftx(x)∈E1∩E2, but this contradicts the assumption E1∩E2=∅). It follows from Morse’s lemma that for any ε∈(0,1) the manifold Mε=φ−1([0,ε]) is homeomorphic to the disjoint union k0=|Ω0ft| of balls of dimension n. By Theorem 3.4 in [33] (also see Theorem 3.2 and the remark to it in [34]) the manifold Qa=φ−1([0,n/2]) is obtained from Mε by attaching k1=|Ω1ft| handles H1,…,Hk1 of index 1 each of which contains precisely one saddle equilibrium, whose Morse index is 1 (Figure 5 shows the phase portrait of the flow ft, the section Σft and the handlebody Qa in the case when g=0 and k1=1). Since Qa is connected, we have k1⩾k0−1. We set ga=k1−k0+1. We claim that Qa is an (n,1)-handlebody of genus ga. There are two cases to consider: ga=0 and ga>0. Let ga=0. We use induction on k0 to show that the compact connected manifold obtained from k0 copies of closed n-balls by attaching (k0−1) handles of index 1 is homeomorphic to the n-ball. This will imply that, in the case ga=0, Qa is homeomorphic to an n-ball (that is, to a handlebody of genus 0). For k0=1 there is one ball and no handles, and in this case the claim is true. Assuming that the claim holds for k0=i⩾1 consider the case k0=i+1. In this case Qa is a union of two balls with an attached handle. The handle can be attached in two ways: in the first case we obtain a disconnected manifold (Figure 6, a), and in the second the resulting manifold is connected (Figure 6, b). Consider the case ga>0. Then Qa is connected and obtained by attaching k1=k0−1+ga handles to k0 balls. According to the above, the connected manifold obtained by attaching k0−1 handles to k0 balls is homeomorphic to the ball. Attaching ga more handles to this manifold we obtain an (n,1)-handlebody of genus ga. So Qa is a handlebody of genus ga=k1−k0+1. In a similar way, considering the flow f−t and its energy function ψ=n−φ, we find that the manifold Qr=Mn∖intQa is an (n,1)-handlebody of genus gr=kn−1−kn+1, where kn=|Ωnft|, kn−1=|Ωn−1ft|. The manifolds Qa and Qr have the common boundary, and so ga=gr. By the definition of the quantities gft, ga and gr, An appeal to Assertion 1 shows that gft=g. Thus, ga=g.Let us prove part 2. Assertion 8 shows that for any point σ1∈Ω1ft the intersection of Wsσ1 and Σft is transversal; hence the set lsσ1=Wsσ1∩Σft is a smooth closed manifold of dimension n−1+n−1−n=n−2. Since Σft is a section for the trajectories of ft, the submanidold lsσ1 is a section for all trajectories of the restriction of the flow ft to the set Wsσ1∖σ1. Consider the retraction that associates with a point x∈Wsσ1∖σ1 the point y=ftx(x)∈lsσ1 of intersection of the set lsσ1 and the trajectory of the flow ft passing through x. The retraction r is connected with the identity map id:Wsσ1∖σ1→Wsσ1∖σ1 via the homothety Therefore, lsσ1 has the same homotopy type as Wsσ1∖σ1. The manifold Wsσ1∖σ1 is homeomorphic to Rn−1∖O, so it has the homotopy type of the (n−2)-sphere. Now, by Poincaré’s theorem the manifold lsσ1 is homeomorphic to the (n−2)-sphere. The arguments for a point σn−1∈Ωn−1ft are similar.Let us prove assertion 3 of Lemma 2. Let g>0. Since Qa is a handlebody of genus g, there exist smooth embeddings ei: [−1,1]×Bn−1→Qa, i∈{1,…,g}, such that:
We set Hi=ei([−1,1]×Bn−1), B−i=ei({−1}×Bn−1) and B+i=ei({1}×Bn−1), i∈{1,…,g}. By the construction of Qa (see the proof of part 1 of Lemma 2) each subset Hi has a nonempty intersection with the stable manifold of at least one saddle equilibrium σi1∈Ω1ft. By Assertion 3 the sphere lsσi1 bounds a ball Bsσi1∈Wsσi1, which contains the point σi1. Therefore, intBsσi1⊂intHi and Bsσi1=Wsσi1∩Hi. The balls {Bsσi1} are smoothly embedded in Qa, and so, for each i∈{1,…,g} there exists a smooth embedding ψi:[−1,1]×Bn−1→Hi with the following properties:
The disc Bsσi1 partitions Hi into two connected components, and therefore cl(Hi∖Ni) consists of two connected components H+i and H−i. It follows from properties 1)–3) that the boundary of each of these components is homeomorphic to a sphere. Indeed, let B+i∪˜B+i⊂H+i. By Assertion 4 the set ∂H+i∖(intB+i∪int˜B+i) (which also lies in the boundary of Hi) is homeomorphic to the annulus Sn−2×[0,1]. Now it follows from Corollary 3 that ∂H+i is homeomorphic to the sphere Sn−1, and an appeal to Assertion 2 shows that H+i is homeomorphic to the ball Bn. A similar analysis shows that H−i is homeomorphic to the ball Bn. By definition cl(Qa∖⋃gi=1Hi) is homeomorphic to the n-ball. Hence the set is also homeomorphic to the n-ball. Setting Tsi=Ni∩Σft, we have therefore, cl(Σft∖⋃gi=1Tsi) is homeomorphic to a sphere from which 2g open discs of dimension n−1 are removed, which is the required result. Lemma 2 is proved.The required set σ1n−1,…,σgn−1∈Ωn−1ft exists since the above arguments also apply to the flow f−t and its energy function n−φ. The following result is a direct consequence of part 3 of Lemma 2. Corollary 5. If g>0, then each of the sets Csft and Cuft contains at least one set {lsσ11,…,lsσg1} or {luσ1n−1,…,luσgn−1} such that for any k∈[1,g] the manifolds (Σft∖⋃ki=1lsσi1) and (Σft∖⋃ki=1luσin−1) are connected. In addition, any g+1 spheres in the set Csft∪Cuft divide Σft into several connected components. Definition 5. The set of saddle equilibria {σ11,…,σg1} ({σ1n−1,…,σgn−1}) described in part 3 of Lemma 2 and the set of spheres {lsσi1} ({luσjn−1}, respectively) corresponding to the intersection of their stable (respectively, unstable) manifolds with the section Σft are called a maximal nonseparating s-set (respectively, u-set). 3.3. The relationship between between a bi-colour graph and the scheme of the flow ft∈G(Sng)Let ηft:V(Γft)∪E(Γft)→Dft∪Lft be a bijection such that ηft(V(Γft))=Dft, ηft(E(Γft))=Lft, and an edge e∈E(Γft) is incident to vertices v,w∈V(Γft) if and only if the sphere ηft(e)∈Lft lies on the boundary of the domains ηft(v),ηft(w)∈Dft. Let dft be the set of connected components of the set Σft∖(Csft∪Cuft). By Lemma 2 each sphere L∈Lft (and therefore each domain D∈Dft) intersects the characteristic section Σft in precisely one connected component. Hence the bijection ηft induces a bijection η∗:V(Γft)∪E(Γft)→dft∪(Csft∪Cuft) with the same properties. In what follows the graph Γft is identified with a one-dimensional polyhedron embedded in the characteristic section Σft as in the following statement (see Figure 8). Proposition 4. The graph Γft embeds in the characteristic section Σft so that to each vertex v∈V(Γft) there corresponds a point ˜v in the domain η∗(v)∈dft, and to an edge e∈E(Γft) joining vertices v,w∈V(Γft) there corresponds a smooth arc ˜e⊂Σft joining the points ˜v and ˜w and intersecting the sphere l=η∗(e)∈Csft∪Cuft in one point. In what follows the point ˜v and the arc ˜e are called a vertex and an edge of the graph Γft, respectively; the symbol ∼ is suppressed in the notation. We say that an edge e separates the graph Γft if the number of connected components of the graph Γft∖e is greater than that of Γft. Proposition 5. 1. The graph Γft is connected. 2. An edge e∈Γft separates the graph Γft if and only if the sphere η∗(e)=l separates the section Σft. Proof. Let us prove claim 1. Let v and w be vertices of the graph Γft. It follows from Lemma 2 that the section Σft is connected. Hence the points v and w can be connected by a path γ:[0,1]→Σft. These points lie in different connected components of the set Σft∖(Csft∪Cuft), and so the image γ([0,1]) of the path γ intersects some spheres l1,…,lk∈Csft∪Cuft and traverses the connected components d1,…,dk+1 of the set Σft∖(Csft∪Cuft). Assume that γ(0)=v∈d1 and γ(1)=w∈dk. Then there is a route M in Γft composed of vertices v1=v,v2,…,vk+1=w and edges e1,…,ek such that vi∈di, and the edge ei intersects the sphere li, i∈{1,…,k} (Figure 9). Hence the graph Γft is connected.
Let us prove claim 2. Let the vertices v and w be incident to an edge e such that Γft∖e is disconnected and l=η∗(e). We claim that the set Σft∖l is disconnected. Assume on the contrary that Σft∖l is connected. Proceeding as in the proof of claim 1 we can construct a route M in Γft that connects the points v and w and does not contain the edge e. The existence of such a route means that the graph Γft∖e is connected, which contradicts the assumption. Therefore, Σft∖l is disconnected. Now assume that Σft∖l is disconnected. We claim that the graph Γft∖e is too. Assume for a contradiction that Γft∖e is connected. Then there exists a route M in Γft∖e that connects the vertices v and w. The route M does not intersect the sphere l, and therefore v and w lie in the same connected component of Σft∖l. The sets Σft∖l and e∖l are disconnected, and so Γft∖l is too, and the vertices v and w incident to the edge e lie in different connected components of the set Γft∖l. This contradiction shows that the edge e cannot lie in a cycle of the graph Γft∖e. Proposition 5 is proved. Lemma 3. Let ft∈G(Sng). If g=0, then Γft is a tree. If g>0, then Γft is connected and contains precisely g simple pairwise distinct cycles such that: 1) no edge lies simultaneously in two cycles; 2) each cycle of the graph Γft contains both an edge of colour s and an edge of colour u, and these edges correspond to spheres lsσi1∈Csft and luσjn−1∈Cuft which lie in maximal nonseparating s- and u-sets, respectively. Proof. Let g=0. By Lemma 2 the characteristic section Σft is a sphere of dimension n−1, and for any sphere l∈Lft the intersection l∩Σft is a sphere of dimension n−2. We recall that μft, the number of saddles in the flow ft, is equal to the number of spheres in the set Lft. By Corollary 1 the set Σft∖⋃l∈Lftl has precisely μft+1 connected components. Therefore, the graph Γft has μft+1 vertices and μft edges. By Proposition 5, Γft is connected. Now, by Assertion 7 the graph Γft is a tree.
Let g>0. By part 3 of Lemma 2 there exists a maximal nonseparating s-set of spheres lsσ11,…,lsσg1∈Csft such that the set ˜Σft=Σft∖⋃gi=1lsσi1 is connected and, for any sphere l∈Csft∪Cuft∖⋃gi=1lsσi1, the set ˜Σft∖l is disconnected. Let es1,…,esg and e be the edges of Γft corresponding to the spheres lsσ11,…,lsσg1 and l, respectively. From Proposition 5 it follows that the graph ˜Γft=Γft∖⋃gi=1esi is connected and ˜Γft∖e is disconnected. Hence ˜Γft is connected and contains no cycles, so that it is a tree. Now it follows from Assertion 7 that each edge es1,…,esg lies on a simple cycle of the graph Γft. No two edges esi, esj, i≠j lie on the same cycle since otherwise Γft∖(esi∪esj) would have two connected components. Further, no edge lies in two cycles simultaneously, since otherwise ˜Γft would be disconnected (Figure 10). So Γft contains at least g pairwise distinct simple cycles, each of which contains an edge esi of colour s. Since ˜Γft is acyclic, Γft contains precisely g cycles. By part 3 of Lemma 2, in addition to the maximal nonseparating s-set, there also exists a maximal nonseparating u-set of spheres luσ1n−1,…,luσgn−1 in the set Cuft. To each sphere luσin−1 in this set there corresponds an edge eui of Γft coloured with colour u. Replacing s by u in the above analysis we find that each of the g simple cycles of Γft, along with the edge esi of colour s, also contains an edge eui of colour u. Lemma 3 is proved. 3.4. The proof of Theorem 1The necessity of the assumptions of the theorem follows directly from the definition of topological equivalence. For sufficiency we show that the existence of an isomorphism ξ between two graphs Γft and Γf′t implies the equivalence of the schemes Sft and Sf′t. With this proviso, Lemma 1 would imply that the flows ft and f′t are topologically equivalent. Let ηft:V(Γft)∪E(Γft)→Dft∪Lft be a bijection such that an edge e∈E(Γft) is incident to vertices v,w∈V(Γft) if and only if the sphere ηft(e)∈Lft lies on the boundary of the domains ηft(v),ηft(w)∈Dft (see § 3.3). We denote by ηf′t:V(Γf′t)∪E(Γf′t)→Df′t∪Lf′t a bijection with the same properties with regard to the flow f′t. The isomorphism ξ:V(Γft)→V(Γf′t) induces an isomorphism ξ∗:Dft→Df′t as follows: ξ∗=ηf′tξη−1ft|Dft. Since ξ preserves adjacency and the colours of edges, the isomorphism ξ∗ extends naturally to an isomorphism between the sets Lft and Lf′t; this isomorphism will also be denoted by ξ∗. Let μft and νft (μf′t and νf′t) be the numbers of saddle and nodal equilibria of the flow ft (of f′t, respectively). Since μft=|Lft|=|E(Γft)|, μf′t=|Lf′t|=|E(Γf′t)| and the graphs Γft and Γf′t are isomorphic, it follows that μft=μf′t. We claim that νft=νf′t. The nodal states of the flow ft can be partitioned into two subsets: points of the first lie on spheres in Lft, and in this case, as the graphs Γft and Γf′t are isomorphic, for ft and f′t these sets have the same cardinality. Points of the second subset lie in domains in the set Dft, each domain D∈Dft containing at most one sink or source equilibrium. In addition, the sink (source) equilibrium ω (α, respectively) lies in a domain D∈Dft if and only if its boundary contains only stable (unstable) separatrices of saddle equilibria, whose Morse index is 1 (n−1, respectively). Therefore, in this case the vertex corresponding to the domain D is incident only to edges of colour s (of colour u). Hence, since the graphs Γft and Γf′t are isomorphic, it follows that νft=νf′t. The type of the ambient manifold of the flow ft is a function of the quantity gft=(μft−νft+2)/2, and therefore gft=gf′t. Next we set Let Σft and Σf′t be the characteristic sections of the flows ft and f′t. By definition, each sphere L∈Lft (L′∈Lf′t) and each domain D∈Dft (D′∈Df′t) intersects Σft (intersects Σf′t, respectively) in one connected component. Hence the isomorphism ξ∗ induces in a natural way a bijection between the elements of Cuft and Cuf′t, between Csft and Csf′t, and also between the connected components of the sets Σft∖(Cuft∪Csft) and Σf′t∖(Cuf′t∪Csf′t), which we also denote by ξ∗. By the definition of ξ∗ a connected component d⊂Σft∖(Cuft∪Csft) corresponds to a connected component d′=ξ∗(d)⊂Σf′t∖(Cuf′t∪Csf′t) if and only if, for any connected component l⊂Cuft∪Csft of the boundary of the domain d, there exists a connected component l′=ξ∗(l)⊂Cuf′t∪Csf′t of the boundary of d′. Now we construct a homeomorphism h:Σft→Σf′t such that h(l)=l′ for any sphere l⊂Csft∪Cuft. There are two cases to consider, g=0 and g>0. Case 1: g=0. By Lemma 3 the graphs Γft and Γf′t are trees, and by Lemma 2 the characteristic sections Σft and Σf′t are homeomorphic to spheres. Let r be the maximum rank of the vertices of Γft and Γf′t. There are two cases to consider: Case 1, a). Let v0 be the central vertex of Γft. If r=0, then Γft consists of the unique vertex v0, and the nonwandering set of the flow ft does not contain saddle equilibria. In this case the nonwandering set of ft consists of precisely two saddle equilibria, a source and a sink, and all such flows are topologically equivalent. Let r>0. Then the vertex v0 has degree δ0⩾2 (if δ0=1, then v0 is a leaf; such a vertex is central only in the case when r=1 and the graph Γft is bicentral). Let v0,1,…,v0,δ0 be the vertices adjacent to v0. We denote by l0,i the sphere in Csft∪Cuft that corresponds to the edge (v0,v0,i). We set l0=⋃δ0i=1l0,i. Then the boundary of the domain d∈Dft∩Σft corresponding to the vertex v0 consists precisely of all spheres in l0 (see Figure 11). We denote the subset consisting of the spheres in Csf′t∪Cuf′t corresponding to edges incident to the central vertex v′0∈V(Γ′ft) by l′0. We assume that the spheres in l′0 are labelled so that l′0,i=ξ∗(l0,i), i∈{1,…,δ0}. By Assertion 3 each sphere l0,i∈l0 partitions Σft into two connected components, and the closure of each component is homeomorphic to the (n−1)-ball. Hence b0=Σft∖cld is a union of pairwise disjoint open balls bounded by spheres in l0. We denote the complement to the set cld′∈Df′t∩Σf′t corresponding to the vertex v′0 by b′0. Since the graphs Γft and Γf′t are isomorphic, b′0 consists of the same number of balls as b0, and for each ball b0,i∈b0 bounded by a sphere l0,i∈l0 there exists a ball b′0,i∈b′0 bounded by a sphere l′0,i∈l′0. It follows from Corollary 4 and Assertion 6 that there exists a homeomorphism h0:Σft→Σf′t such that h0(l0,i)=l′0,i for each i∈{1,…,δ0}. If r=1, then h0 is the required homeomorphism. Let r>1. Then the set Cuft∪Csft∖l0 is nonempty, and all spheres in this set lie in b0. Since the bi-colour graphs are isomorphic, the set Csf′t∪Csf′t∖l′0 is nonempty and all spheres in this set lie in b′0. Let δi be the degree of the vertex v0,i. Since r>1, we have δi>1 for any i∈{1,…,δ0}. Let vi,j be a vertex of Γft adjacent to v0,i and distinct from v0, j∈{1,…,δi}. Let li,j be the sphere in Cuft∪Csft∖l0 corresponding to the edge (vi,vi,j), and let l′i,j be the sphere in Cuf′t∪Csf′t∖l′0 such that l′i,j=ξ∗(li,j). We set l1,i=⋃δij=1li,j and l′1,i=⋃δij=1l′i,j, i∈{1,…,δ}. By the definition of the graph Γft the spheres in l1,i and l0,i form the boundary of some domain di⊂Dft∩Σft corresponding to the vertex vi and contained in the ball b0,i⊂b0 bounded by the sphere l0,i (see Figure 11). We denote by b1,i the set of balls lying in b0,i and bounded by spheres in l1,i. We have h0(b1,i)⊂b′0,i. By Proposition 3, for each i∈{1,…,δ0} there exists a homeomorphism hi:Σf′t→Σf′t such that hi|Σf′t∖intb′0,i=id and hi(h0(l1,i))=l′1,i for any sphere l1,i∈l1,i. If r=2, then the composition hδhδ−1⋯h1h0 is the required homeomorphism h:Σft→Σf′t. Otherwise, we continue the process and, after a finite number of steps, arrive at the required homeomorphism. Case 1, b). The tree Γft is bicentral. We denote the sphere corresponding to the central edge of the graph Γft by l0. Let b0 be an arbitrary connected component of Σft∖l0, and let v be the central vertex corresponding to the domain d⊂Dft∩Σft lying in b0 (then the boundary of d contains the sphere l0). We denote the sphere corresponding to the central edge of the graph Γf′t by l′0. Let b′0 be the connected component of the set Σf′t∖l′0 that contains the domain d′. By Assertion 6 there exists a homeomorphism h0:Σft→Σf′t such that h0(l0)=l′0 and h0(b0)=b′0. If r=0, then h0 is the required homeomorphism. If r>0, then we continue the process of constructing the homeomorphism h as in case 1, a). Case 2: g>0. By part 3 of Lemma 2 there exist precisely g spheres lsσ11,…,lsσg1∈Csft such that the set Σft∖⋃gi=1lsσi1 is connected. By Lemma 3 the edges es1,…,esg corresponding to lsσ11,…,lsσg1 lie in pairwise different cycles of the graph Γft. We denote the vertices of Γft incident to the edge esi, i∈{1,…,g}, by vi and wi. Since the graphs Γft and Γf′t are isomorphic, the edges e′s1=ξ(es1), …, e′sg=ξ(esg) lie in pairwise different cycles of Γf′t. By Proposition 5 the set of spheres l′sσ′11=ξ∗(lsσ11), …, l′sσ′g1=ξ∗(lsσg1) corresponding to these edges does not separate the section Σf′t. We denote the sets of pairwise disjoint tubular neighbourhoods of spheres in the sets Csft∪Cuft and Csf′t∪Cuf′t by {Tl,l∈Csft∪Cuft} and {Tl′,l′∈Csf′t∪Cuf′t}, respectively. By part 3 of Lemma 2, both Σft∖⋃gi=1intTlsσi1 and Σf′t∖⋃gi=1intTl′sσ′i1 are homeomorphic to a sphere of dimension n−1 with 2g holes. To the manifold Σft∖⋃gi=1intTlsσi1 we attach the set B consisting of 2g copies of the ball Bn−1. Let ˆΣft be the sphere thus obtained. Let pft: Σft∖⋃gi=1intTlsσi1∪B→ˆΣft be the natural projection. In what follows the sets pft(Σft∖⋃gi=1intTlsσi1), pft(Csft), pft(Cuft), p_{f^t}(\partial T_{l^{\mathrm s}_{\sigma_1^i}}) and p_{f^t}(\mathfrak B) are denoted by \Sigma_{{f}^t}\setminus \bigcup _{i=1}^g \operatorname{int} T_{l^{\mathrm s}_{\sigma_1^i}}, C^{\mathrm s}_{f^t}, C^{\mathrm u}_{f^t}, \partial T_{l^{\mathrm s}_{\sigma_1^i}} and \mathfrak B, respectively. We remove the edges e_1, \dots, e_g from \Gamma_{f^t}, and complement each vertex v_i and w_i with vertices v_{i+} and w_{i+} and the edges (v_i, v_{i+}) and (w_i, w_{i+}) of colour \mathrm s. The resulting tree is denoted by \widehat{\Gamma}_{f^t}. With each new vertex v_{i+} and w_{i+} we associate a ball added to the manifold \Sigma_{{f}^t}\setminus \bigcup _{i=1}^g \operatorname{int} T_{l_i}, and with each new edge we associate the boundary of this ball. We denote the set of spheres obtained as the union of C^{\mathrm s}_{f^t}\setminus \bigcup_{i=1}^g l^{\mathrm s}_{\sigma_1^i} and the set of boundaries of the added balls by \widehat{C}^{\mathrm s}_{f^t}. There is a natural bijection \widehat{\eta}_{f^t} between the edge set of the graph \widehat{\Gamma}_{f^t} and the elements of \widehat{C}^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t}; there is also a bijection between the edge set of the graph \widehat{\Gamma}_{f^t} and the connected components of the set \widehat{\Sigma}_{{f}^t}\setminus (\widehat{C}^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t}). We proceed similarly with the flow {f'}^t, and denote by \widehat{\Gamma}_{{f'}^t}, \widehat{\Sigma}_{{f'}^t}, \widehat{C}^{\mathrm s}_{{f'}^t} and \mathfrak B' the resulting objects corresponding to the analogous objects for f^t. The isomorphism \xi\colon \Gamma_{f^t}\to \Gamma_{{f'}^t} extends naturally to an isomorphism \widehat{\xi}\colon \widehat{\Gamma}_{f^t}\to \widehat{\Gamma}_{{f'}^t}, which induces a bijection \widehat\xi_* between the elements of the sets \widehat{C}^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t} and \widehat{C}^{\mathrm s}_{{f'}^t}\cup C^{\mathrm u}_{{f'}^t}. Proceeding in accordance with the algorithm developed for g=0, we construct a homeomorphism \widehat{h}\colon \widehat{\Sigma}_{f^t}\to \widehat{\Sigma}_{{f'}^t} such that \widehat{h}(l)= \widehat \xi_*(l) for any sphere l\in \widehat{C}^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t}. The homeomorphism \widehat h induces naturally a homeomorphism between the sets \Sigma_{{f}^t}\setminus \bigcup _{i=1}^g \operatorname{int} T_{l_i} and \Sigma_{{f'}^t}\setminus \bigcup _{i=1}^g \operatorname{int} T_{l'_i}, which we also denote ny \widehat h. By Proposition 2, \widehat{h} extends to each annulus T_{l^{\mathrm s}_{\sigma_1^i}}, i\in \{1,\dots,g\} as a homeomorphism h\colon \Sigma_{{f}^t}\to \Sigma_{{f'}^t} such that h|_{\Sigma_{{f}^t}\setminus \bigcup _{i=1}^g \operatorname{int} T_{l^{\mathrm s}_{\sigma_1^i}}}=\widehat h and h(l^{\mathrm s}_{\sigma_1^i})={l'}^{\mathrm s}_{{\sigma'}_1^i} for any i\in \{1,\dots,g\}. So h is the required homeomorphism. Theorem 1 is proved. 3.5. A realization of classes of topological equivalenceDefinition 6. A simple connected graph \Gamma whose edges are coloured with colours \mathrm s and \mathrm u is called admissible if it has g\geqslant 0 pairwise distinct simple cycles, each cycle has at least one edge of colour \mathrm s and at least one edge of colour \mathrm u, and no edge lies simultaneously in two cycles. Proof. By definition the graph \Gamma_{f^t} contains no loops, that is, \Gamma_{f^t} is simple. The remaining conditions in the definition of an admissible graph are satisfied for \Gamma_{f^t} by Lemma 3. Proposition 6 is proved. Lemma 4. Let \Gamma_{f^t} be a bi-colour graph of a flow f^t\in G(\mathcal{S}^n_g), v, w\in V(\Gamma_{f^t}) be vertices connected by a unique simple route \mathcal{M}, and let d_v and d_w be the connected components of the set \mathcal{D}_{f^t} corresponding to v and w. If \mathcal{M} contains an edge e^{\mathrm s} of colour \mathrm s, then the closures of the sets d_v and d_w contain two nonidentical sink equilibria \omega_v and \omega_w of the flow {f}^t. Proof. We assume that the graph \Gamma_{f^t} is embedded in the manifold \mathcal{S}^n_{g} as in Proposition 4.
Let L^{\mathrm s} be the sphere in \mathcal{L}_{f^t} corresponding to the edge e^{\mathrm s} and \sigma\in \Omega^1_{f^t} be an equilibrium such that L^{\mathrm s}=\operatorname{cl} W^{\mathrm s}_\sigma. By assumption \mathcal{M} is the unique simple route connecting v and w, and so the edge e^{\mathrm s} does not lie in any cycle of \Gamma_{f^t}. By Proposition 5 the sphere L^{\mathrm s}\subset\mathcal{L}_{{f}^t} divides the manifold \mathcal{S}^n_{g} into two connected components V and W. The route \mathcal{M} is also divided by L^{\mathrm s} into two connected components so that the end vertices v and w lie in different components. Hence the domains d_v and d_w also lie in different connected components among V and W. Let d_v\subset V and d_w\subset W. There are two cases to consider:
In case 1) let e^{\mathrm u}_v and e^{\mathrm u}_w be the edges incident to the vertices v and w, respectively, and lying in the route \mathcal{M}. By assumption they have colour \mathrm u. Hence the spheres L^{\mathrm u}_v, L^{\mathrm u}_w\subset \mathcal L_{f^t} corresponding to e^{\mathrm u}_v and e^{\mathrm u}_w lie in the sets \operatorname{cl} d_v and \operatorname{cl} d_w, respectively. By the definition of \mathcal{L}_{f^t} there exists sinks \omega_v\in L^{\mathrm u}_v and \omega_w\in L^{\mathrm u}_w. The spheres L^{\mathrm u}_v and L^{\mathrm u}_w are disjoint from the sphere L^{\mathrm s}, and therefore lie in different connected components among V and W. Hence the sinks \omega_v and \omega_w are different. Consider case 2). The one-dimensional unstable separatrices of the point \sigma\in \Omega^1_{{f}^t} such that L^{\mathrm s}= \operatorname{cl} W^{\mathrm s}_{\sigma} also lie in distinct sets among V and W. Therefore, there exist sinks \omega_+\subset V and \omega_-\subset W lying in the closure of the set W^{\mathrm u}_{\sigma}. There are three cases to consider:
In case (a) L^{\mathrm s}\subset \operatorname{cl} d_v and L^{\mathrm s}\subset \operatorname{cl} d_w, and therefore W^{\mathrm u}_\sigma\cap d_v\neq \varnothing, W^{\mathrm u}_\sigma\cap d_w\neq \varnothing and \omega_+ \subset \operatorname{cl} d_v, \omega_-\subset \operatorname{cl} d_w. We set \omega_v=\omega_+ and \omega_u=\omega_-. In case (b) we assume for definiteness that the edge e^{\mathrm s} is incident to the vertex v. Then there exists an edge e^{\mathrm u}\in \mathcal{M} incident to w and of colour \mathrm u. In this case L^{\mathrm s}\subset \operatorname{cl} d_v, and so W^{\mathrm u}_\sigma\cap d_v\neq \varnothing and \omega_+\subset \operatorname{cl} d_v\cap W. In addition, there exists a sink equilibrium \omega lying on the sphere L^{\mathrm u}\in \mathcal{L}_{{f}^t} and corresponding to e^{\mathrm u}. Hence L^{\mathrm u}\subset \operatorname{cl} d_w, and therefore \omega\subset \operatorname{cl} d_w. Since L^{\mathrm s}\cap L^{\mathrm u}=\varnothing, we have \omega_+\neq \omega. We set \omega_v=\omega_+ and \omega_w=\omega. In case (c) we denote by e^{\mathrm s}_v and e^{\mathrm s}_w two edges lying in the route \mathcal{M} and incident to the vertices v and w, respectively. Both e^{\mathrm s}_v and e^{\mathrm s}_w are of colour \mathrm s. We denote by L^{\mathrm s}_v and L^{\mathrm s}_w the spheres in \mathcal{L}_{f^t} corresponding to e^{\mathrm s}_v and e^{\mathrm s}_w, respectively, and denote by \sigma_v, \sigma_w\in \Omega^1_{f^t} saddle equilibria such that L^{\mathrm s}_v=W^{\mathrm s}_{\sigma_v} and L^{\mathrm s}_w=W^{\mathrm s}_{\sigma_w}. The union L^{\mathrm s}_v\cup L^{\mathrm s}_w divides the manifold \mathcal{S}^{n}_{g} into three connected components W, V and U, each of which contains at least one sink lying in the closure of the set W^{\mathrm u}_{\sigma_v}\cup W^{\mathrm u}_{\sigma_w}. We denote these sink points by \omega_w, \omega_v and \omega_u, respectively. Let d_w\subset W and d_v\subset V. Then \omega_v\subset \operatorname{cl} d_v, \omega_w\subset \operatorname{cl} d_w and \omega_v\neq \omega_w. Lemma 4 is proved. Theorem 2. For any admissible graph \Gamma there exists a flow f^t\in G(\mathcal{S}^n_g) whose graph \Gamma_{f^t} is isomorphic to the graph \Gamma by an isomorphism preserving the colours of the edges. Proof. We proceed by induction on g. For g=0 the graph \Gamma is a tree, and an algorithm for the realization of \Gamma by a G(\mathcal{S}^n_0)-flow was described in [35]. In particular, it was shown there how to construct a flow f^t_0 on the sphere \mathcal S^n_0 whose nonwandering set consists of precisely four equilibria: two sources, one sink and a saddle of index n-1. The phase portrait of the flow f^t_0 and its bi-colour graph are shown in Figure 12.
Let \psi\colon \mathcal S^n_0\to [0,n] be the energy function of f^t_0. We set N=\psi^{-1}[0, n-0.5]. Then N is a manifold with boundary obtained from \mathcal S^n_0 by removing two disjoint open balls with smoothly embedded boundaries. By Assertion 4 the manifold N is homeomorphic to \mathbb{S}^{n-1}\times [-1,1]. By the definition of the energy function the trajectories of f^t_0 are transversal to the boundary of N. Assume that for any admissible graph with i\in \{0,1,\dots,g-1\} simple cycles we have constructed a G(\mathcal{S}^n_i)-flow whose graph is isomorphic to this admissible graph by means of an isomorphism preserving the colours of edges. Let us construct a flow f^t\in G(\mathcal{S}^n_g) for an admissible graph \Gamma with precisely g>0 cycles each of which has pairwise distinct vertices. Let (v,w) be an edge of \Gamma lying in a cycle. For definiteness we assume that this edge has colour \mathrm u (the arguments for the colour \mathrm s are similar). By the induction assumption the graph \Gamma_* obtained by removing the edge (v,w) from \Gamma is realized by a flow f_*^t\in G(\mathcal{S}^n_{g-1}) such that the graph \Gamma_{f_*^t} is isomorphic to \Gamma_*. Since the vertices v and w lie in a cycle on \Gamma, the graph \Gamma_* has a unique simple route \mathcal{M} connecting v and w and containing an edge e^{\mathrm s} of colour \mathrm s. By Lemma 4 the closures of the domains d_v, d_w\subset \mathcal{D}_{f_*^t} corresponding to v and w contain two nonidentical sink equilibria \omega_v and \omega_w of {f}^t. Let \varphi\colon \mathcal{S}^n_{g-1}\to [0,n] be the energy function of the flow f_*^t. We denote by B_v and B_w the connected components of the set \varphi^{-1}([0, 0.5]) containing the points \omega_v and \omega_w, respectively. We remove the interiors of the balls B_v and B_w from the manifold \mathcal{S}^n_{g-1}, and to the resulting manifold with boundary we attach the manifold N homeomorphic to the annulus \mathbb{S}^{n-1}\times [-1,1] equipped with the model flow f^t_{0}. Let M^n be the manifold obtained from \mathcal{S}^n_{g-1} by removing the interiors of B_v and B_w and attaching N to the manifold with boundary thus obtained by means of a diffeomorphism h\colon \partial (B_v\cup B_w)\to \partial N such that h(\mathcal{L}_{{f'}^t})\cap W^{\mathrm u}_{\sigma_0}=\varnothing, where \sigma_0 is a saddle equilibrium of f^t_0. According to [36], the manifold M^n is homeomorphic to \mathcal S^n_g. Let p: \mathcal{S}^n_{g-1}\setminus \operatorname{int} (B_v\cup B_w)\cup N\to M^n denote the natural projection. By smoothing the flow f^t_0 near the boundary of N we define a flow f^t on M^n which coincides with f_*^t on the set p(\mathcal{S}^n_{g-1}\setminus (B_v\cup B_w)) and agrees with f^t_0 on p(N). By construction the bi-colour graph \Gamma_{f^t} of f^t can be obtained from \Gamma_* by adding to \Gamma_* an edge of colour \mathrm u connecting v and w (see Figure 12). Therefore, the graphs \Gamma_{f^t} and \Gamma are isomorphic. Theorem 2 is proved. 
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\Bibitem{GriGur23}
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