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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
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 5, DOI: https://doi.org/10.4213/sm9761 
Bibliographic databases:
    § 1. Introduction and statement of the resultsWe recall that a flow $f^t$ on a smooth closed manifold $M^n$ 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 $c_i$ be the number of equilibria such that the dimension of their unstable manifolds (Morse index) is $i\in \{0,\dots, n\}$, and let $\chi(M^n)$ be the Euler characteristic of the manifold $M^n$. Then Note that by (1.1) the number $|\chi(M^n)|$ is a lower bound for the number of saddle equilibria. If $n=2$ and $M^2$ is orientable, then, as is known, $M^2$ is homeomorphic to the connected sum
$$
\begin{equation*}
\mathbb{S}^2\,\sharp\,\mathbb{T}^2\,\sharp\,\dotsb \,\sharp\, \mathbb{T}^2
\end{equation*}
\notag
$$
of the 2-sphere $\mathbb{S}^2$ and $g\geqslant 0$ tori $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1$; it is also known that, $\chi(M^2)=2-2g$. In this case it follows from (1.1) that the genus $g$ of the manifold $M^2$ can be expressed in terms of the number of nodal equilibria $\nu_{f^t}=c_0+c_2$ and the number of saddle equilibria $\mu_{f^t}=c_1$ as follows: For $n>2$ the Euler characteristics is no longer a complete topological invariant; moreover, if $n$ is odd, then $\chi(M^n)=0$ for any manifold $M^n$, 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 $G(M^n)$ be the class of gradient-like flows defined on a closed connected orientable manifold $M^n$ of dimension $n\geqslant 3$ such that, for any $f^t\in G(M^n)$, the invariant manifolds of different saddle equilibria are disjoint. We denote by $\mathcal{S}^n_g$ the manifold homeomorphic to the connected sum
$$
\begin{equation*}
\mathbb{S}^n\, \sharp\, \mathbb{S}^{n-1}\times \mathbb{S}^1\, \sharp\, \dotsb\, \sharp \, \mathbb{S}^{n-1}\times \mathbb{S}^1
\end{equation*}
\notag
$$
of the sphere $\mathbb{S}^n$ and $g$ copies of the direct product $\mathbb{S}^{n-1}\times \mathbb{S}^1$. Let $f^t\in G(M^n)$, $n\geqslant 3$, and let $\nu_{f^t}$, $\mu_{f^t}$ be, respectively, the number of nodal and saddle equilibria of the flow $f^t$. We also set $g_{f^t}=(\mu_{f^t}-\nu_{f^t}+2)/2$. The following result holds. Assertion 1. 1. If the Morse index of each saddle equilibrium of a flow $f^t$ is $1$ or $n-1$, then $g_{f^t}$ is a positive integer, and the ambient manifold $M^n$ is $\mathcal{S}^n_{g_{f^t}}$. 2. If the ambient manifold $M^n$ is $\mathcal{S}^n_{g}$, then the Morse index of each saddle equilibrium of $f^t$ is equal to $1$ or $n-1$ and, in addition, $g=g_{f^t}$. 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 $G(S^n)$-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 $f^t$ 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 $G(\mathcal{S}^n_g)$-flows for $g>0$ and $n\geqslant 3$. For $n=3$ a topological classification of $G(\mathcal{S}^3_g)$-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 $G(\mathcal{S}^n_g)$-flows in the case $g>0$ (for any $n\geqslant 3$). In Figure 2 we show the phase portraits and phase diagrams of two $G(\mathcal{S}^n_1)$-flows. The Morse index of the saddle equilibria $\sigma_1$, $\sigma_3$ and $\sigma_4$ ($\sigma'_1$, $\sigma'_3$ and $\sigma'_4$) is $1$, the Morse index of the saddle $\sigma_2$ (respectively, $\sigma'_2$) is $n - 1$. The union $\operatorname{cl} W^{\mathrm s}_{\sigma_1}\cup \operatorname{cl} W^{\mathrm u}_{\sigma_2}$ ($\operatorname{cl} W^{\mathrm s}_{\sigma'_1}\cup \operatorname{cl} W^{\mathrm u}_{\sigma'_2}$) of the stable and unstable manifolds of $\sigma_1$ and $\sigma_2$ ($\sigma'_1$ and $\sigma'_2)$, respectively, divides the ambient manifold $\mathcal{S}^n_1=\mathbb S^{n-1}\times \mathbb S^1$ into two connected components $D_1$ and $D_2$ (respectively, $D'_1$ and $D'_2$). The equilibria $\sigma_3$ and $\sigma_4$ of the flow $f^t$ lie in the same connected component $D_1$, and the saddle equilibria $\sigma'_3$ and $\sigma'_4$ of ${f'}^t$ lie in different connected components $D_1'$ and $D_2'$, respectively. Hence there exists no homeomorphism of $\mathcal{S}^n_1$ onto itself that maps trajectories of $f^t$ to those of ${f'}^t$. We show below that a classification of $G(\mathcal{S}^n_g)$-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 $S^n$ of dimension $n\geqslant 4$. Let $\Omega^i_{f^t}$ be the set of saddle equilibria of the flow $f^t$ whose Morse index is $i\in \{0,1, n-1, n\}$. Let $f^t\in G(\mathcal{S}^n_g)$ and $\sigma_1\in \Omega^1_{f^t}$ ($\sigma_{n-1}\in \Omega^{n-1}_{f^t}$). It follows from Theorem 2.3 in [13] that the closure of a stable (respectively, unstable) manifold of a saddle equilibrium of the flow $f^t$ contains, in addition to the manifold itself, a unique point, which is a source (sink) equilibrium. Hence the set $L^{\mathrm s}_{\sigma_1}=\operatorname{cl}W^{\mathrm s}_{\sigma_1}$ ($L^{\mathrm u}_{\sigma_{n-1}}=\operatorname{cl}W^{\mathrm u}_{\sigma_{n-1}}$) is a sphere of dimension $n-1$. We denote the set of all spheres $\{L^{\mathrm s}_{\sigma_1}, L^{\mathrm u}_{\sigma_{n-1}}, \sigma_1\in \Omega_{f^t}^1, \sigma_{n-1}\in \Omega_{f^t}^{n-1}\}$ by $\mathcal{L}_{f^t}$ and the set of all connected components of the manifold $\mathcal{S}^n_g\setminus \bigl(\bigcup_{\sigma_1\in \Omega^{1}_{f^t}} L^{\mathrm s}_{\sigma_1} \cup \bigcup _{\sigma_{n-1}\in \Omega^{n-1}_{f^t}} L^{\mathrm u}_{\sigma_{n-1}}\bigr)$ by $\mathcal{D}_{f^t}$. Definition 1. The bi-colour graph of a flow $f^t\in G(\mathcal{S}^n_g)$ is the graph $\Gamma_{f^t}$ with the following properties:
Definition 2. Graphs $\Gamma_{f^t}, \Gamma_{{f'}^t}$ of flows $f^t,{f'}^t\in G(\mathcal{S}^n_g)$ are isomorphic if there exists a bijection $\zeta\colon V(\Gamma_{f^t})\to V(\Gamma_{{f'}^t})$ 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 $f^t$ and ${f'}^t$. Theorem 1. Two flows $f^t\in G(\mathcal{S}^n_g)$ and ${f'}^t\in G(\mathcal{S}^n_{g'})$ are topologically equivalent if and only if their graphs $\Gamma_{f^t}$ and $\Gamma_{{f'}^t}$ are isomorphic. § 2. Auxiliary results2.1. Embeddings of spheres in a manifold and extensions of local homeomorphismsWe set
$$
\begin{equation*}
\mathbb{B}^{n}=\{(x_1,\dots, x_n)\in \mathbb{R}^n\mid x_1^2+\dots + x_n^2\leqslant 1\}.
\end{equation*}
\notag
$$
By a ball or a disc of dimension $n\geqslant 1$ we mean a manifold $B^n$ homeomorphic to $\mathbb{B}^n$. A sphere of dimension $n-1$ (an open ball of dimension $n$) is a manifold homeomorphic to the boundary $\mathbb{S}^{n-1}$ (the interior $\operatorname{int} \mathbb{B}^n$) of the standard ball $\mathbb{B}^n$. A continuous map $f\colon X\to M^n$ is a topological embedding if $f\colon X\to f(X)$ is a homeomorphism (here $f(X)$ is equipped with the topology induced from $M^n$). The image $f(X)$ is called a topologically embedded manifold. Let $f\colon \mathbb{S}^{n-1}\to M^n$ be a topological embedding. The sphere $S^{n-1}=f(\mathbb{S}^{n-1})$ is called locally flat if, for any point $x\in S^{n-1}$, there exist a neighbourhood $U_x\subset M^n$ and a homeomorphism $\psi\colon U_x\to \mathbb{R}^n$ such that $\psi ({S^{n-1}}\cap U_x)$ is a linear subspace of $\mathbb{R}^n$ 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 $\varphi\colon \mathbb{S}^{n-1}\,{\to}\, S^n$ be a topological embedding, $S^{n-1}=\varphi(\mathbb{S}^{n-1})$, $n>0$. Then the set $S^n\setminus S^{n-1}$ consists of two connected components. Corollary 1. Let $S^{n-1}_1, \dots, S^{n-1}_m\subset {S}^n$, $m\geqslant 1$, be pairwise disjoint topologically embedded spheres,. Then the set ${S}^n\setminus \bigcup _{i=1}^m S^{n-1}_i$ 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 $m\in \{1,\dots,i\}$, let us verify it for $m=i+1$.
Since $m$ is finite, there exists $j\in \{1,\dots, m\}$ such that the sphere $S^{n-1}_j$ divides ${S}^{n}$ into two connected components $V_j$ and $W_j$ such that all spheres in the set $\mathcal L= \bigcup _{k=1}^m S^{n-1}_k\setminus S^{n-1}_j$ lie in the same component, which we denote by $V_j$. By the induction assumption the set ${S}^n\setminus \mathcal L$ consists of $(m-1)+1=m$ components $X_1,\dots, X_{m}$. The set $W_j\cup S^{n-1}_j=\operatorname{cl} W_j$ is connected, and thus it lies fully in one of the connected components of $\mathbb{S}^n\setminus \mathcal L$. We denote this component by $X_{m}$. The set $X_{m}\setminus S^{n-1}_j$ has precisely two connected components $V_j \cap X_{m}$ and $W_j$. Therefore, $\mathbb{S}^n\,{\setminus}\, \bigcup _{k=1}^m S^{n-1}_k$ has precisely $m+1$ connected components $X_1, \dots, X_{m-1}$, $V_j\cap X_{m}$ and $W_{j}$. We recall that Schoenflies’s theorem (see [16] and [17]) asserts that any simple curve on the plane $\mathbb{R}^{2}$ (or the sphere ${S}^2$) is the boundary of a 2-disc in $\mathbb{R}^{2}$ (in $S^{2}$). The following result holds (see [18]). Assertion 3 (generalized Schoenflies theorem). If the sphere $S^{n-1}$ is locally flatly embedded in $S^n$, $n \geqslant 3$, then the closures of the connected components of the complement to $S^{n-1}$ are $n$-balls. Assertion 4 (Annulus Theorem). Let $S_{0}^{n-1}$ and $S^{n-1}_{1}$ be disjoint locally flat ${(n-1)}$-spheres in $S^{n}$, and let $K^{n}$ be an open domain in $S^{n}$ bounded by $S_{0}^{n-1}$ and $S^{n-1}_{1}$. Then the closure of the domain $K^{n}$ is homeomorphic to the annulus $\mathbb{S}^{n-1}\times [0,1]$. 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 $\mathrm{SHC}_n$ on stable homeomorphisms of the sphere ${S}^n$. A homeomorphism $f\colon {S}^n\to {S}^n$ is called stable if it can be represented as a finite composition of homeomorphisms each of which is identical on some open set. Conjecture $\mathrm{SHC}_n$ (now verified; see the references below) claims that each orientation-preserving homeomorphism of the sphere ${S}^n$ is stable. Brown and Gluck proved that Conjecture $\mathrm{SHC}_n$ implies the Annulus Theorem in dimension $n$, and, in its turn, the Annulus Theorem in dimensions $k\leqslant n$ implies Conjecture $\mathrm{SHC}_n$. For $n\leqslant 3$ Conjecture $\mathrm{SHC}_n$ 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 $\mathrm{SHC}_n$ 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 $\mathrm{SHC}_4$. The following result is a consequence of Conjecture $\mathrm{SHC}_n$ (see § 4 in [19]). Assertion 5. Any orientation-preserving homeomorphism of the sphere $ S^{n}$, $n\geqslant 1$, is isotopic to the identity.1 Let $M$ and $N$ be $n$-dimensional closed orientable manifolds with boundaries, $n\geqslant 1$, $X\subset \partial M$ and $Y\subset \partial N$ be closed homeomorphic submanifolds of dimension $n-1$, and $g\colon X\to Y$ be a homeomorphism reversing the natural orientation of the boundary. Consider an equivalence relation on the union $M\cup N$: if ${x\in M\cup N\setminus (X\cup Y)}$, then $x\sim x$; if $x\in X$ and $y\in Y$, then $x\sim g(x)$ and $y\sim g^{-1}(y)$. 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 $g\colon X\to Y$.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 $B^n_1$ and $B^n_2$ be balls of dimension $n$, and let $h\colon \partial B^n_1 \to \partial B^n_2$ be an arbitrary homeomorphism. Then there exists a homeomorphism $H\colon B^n_1\to B^n_2$ such that $H|_{\partial B^n_1}=h|_{\partial B^n_1}$. Proof. Let $h_1\colon B^n_1\to \mathbb{B}^n$ and $h_2\colon B^n_2\to \mathbb{B}^n$ be arbitrary homeomorphisms. Let the homeomorphism $\widetilde h\colon \partial \mathbb{B}^n\,{\to}\, \partial \mathbb{B}^n$ be defined by $\widetilde{h}=h_2 h h_1^{-1}|_{\partial \mathbb B^n}$. We also consider the homeomorphism $\widetilde{H}\colon \mathbb{B}^n\to \mathbb{B}^n$ defined by $\widetilde{H}(rx)=r\widetilde{h}(x)$ for each radius vector $x\in \partial \mathbb{B}^n$ and $r\in [0,1]$. Now the required homeomorphism $H$ is given by the formula $H=h_2^{-1}\widetilde{H}h_1$, which completes the proof. Corollary 2. Let $ B^n_1$ and $B^n_2$ be two balls of dimension $n\geqslant 1$, $g\colon \partial B^n_1\to \partial B^n_2$ be a homeomorphism changing the natural orientation of the boundary and $M^n$ be the manifold obtained from the union $ B^n_1\cup B^n_2$ by gluing by means of $g$. Then $M^n$ is homeomorphic to the sphere $\mathbb S^n$. Proof. Let $D^n_1=\{(x_1,\dots, x_{n+1})\in \mathbb{S}^n\mid x_{n+1}\geqslant 0\}$ and $D^n_2=\{(x_1,\dots, x_{n+1})\in \mathbb{S}^n\mid x_{n+1}\leqslant 0\}$, and let $h_1\colon B^n_1\to D^n_1$ be an arbitrary homeomorphism. By Assertion 6 there exists a homeomorphism $h_2\colon B^n_2\to D^n_2$ such that $h_2|_{\partial B^n_2}=h_1g^{-1}|_{\partial B^n_2}$. Consider the continuous map $H\colon B^n_1\cup B^n_2\to \mathbb{S}^n$ defined by The map $H$ is a homeomorphism on $\operatorname{int} B^n_1\cup \operatorname{int} B^n_2$ such that $H(x)=h_1(x)=h_2(g(x))=H(g(x))$ for all points $x\in \partial B^n_1$ and $g(x)\in \partial B^n_2$. Therefore, $H$ induces a homeomorphism $B^n_1\cup_g B^n_2\to \mathbb{S}^{n}$. 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 $X\times [0,1]$ and such that $M\cap N=\partial M\cap \partial N=X$. Then the manifold $M\cup N$ is homeomorphic to $M$. Proof. By Theorem 2 in [28] there exists a topological embedding $h_0\colon X\times [0,1]\,{\to}\, M$ such that $h_0(X\times \{1\})=X$. We set $M_0=h_0(X\times [0,1])$. Let $h_1\colon X\times [0,1]\to N$ be a homeomorphism such that $h_1(X\times \{0\})=X=h_0(X\times \{1\})$.
To complete the proof it suffices to consider the homeomorphisms $g\colon X\times [0,1]\to X\times [0,1]$ and $\widetilde{h}_1\colon X\times [0,1]\to N$, $h\colon X\times [0,1]\to M_0\cup N$ defined by
$$
\begin{equation*}
g(x,t)=(h_1^{-1}(h_0(x,1)),t), \qquad \widetilde{h}_1=h_1g,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
h(x,t)=\begin{cases} h_0(x,2t),&t\in\biggl[0,\dfrac12\biggr], \\ \widetilde{h}_1(x,2t-1),&t\in\biggl(\dfrac12,1\biggr], \end{cases}
\end{equation*}
\notag
$$
respectively, and define the homeomorphism $H\colon M\cup N\to M$ by
$$
\begin{equation*}
H(x)=\begin{cases} h_0(h^{-1}(x)),&x\in Y=M_0\cup N, \\ x,&x\in M\setminus M_0. \end{cases}
\end{equation*}
\notag
$$
Proposition 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 $B^n_+$ and $B^n_-$ to the annulus $\mathbb{S}^{n-1}\times [0,1]$ is homeomorphic to the sphere $\mathbb{S}^n$. Proposition 2 (extension of a homeomorphism from the boundary to the interior of the annulus). Let $K^{n}=\mathbb{S}^{n-1}\times [0,1]$, and let $\psi_{0}\colon \mathbb{S}^{n-1}\times \{0\}\to \mathbb{S}^{n-1}\times \{0\}$ and $\psi_{1}\colon \mathbb{S}^{n-1}\times \{1\}\to \mathbb{S}^{n-1}\times \{1\}$ be orientation-preserving homeomorphisms. Then there exists a homeomorphism $\Psi\colon K^{n}\to K^{n}$ such that 1) $\Psi|_{\mathbb{S}^{n-1}\times \{0\}}=\psi_{0}|_{\mathbb{S}^{n-1}\times \{0\}}$, $\Psi|_{\mathbb{S}^{n-1}\times \{1\}}=\psi_{1}|_{\mathbb{S}^{n-1}\times \{1\}}$; 2) $\Psi(\mathbb{S}^{n-1}\times \{1/2\})=\mathbb{S}^{n-1}\times \{1/2\}$. Proof. By Assertion 5 there exist isotopies $H_{0}\colon \mathbb S^{n-1}\times [0,1] \to \mathbb S^{n-1}\times [0,1]$ and $H_{1}\colon \mathbb S^{n-1}\times [0,1] \to \mathbb S^{n-1}\times [0,1]$ such that
1) $H_{0}|_{\mathbb S^{n-1}\times \{0\}}=\psi_{0}$, $H_{0}|_{\mathbb S^{n-1}\times \{1\}}=\mathrm{id}$; 2) $H_{1}|_{\mathbb S^{n-1}\times \{ 0 \}}=\psi_1$, $H_{1}|_{\mathbb S^{n-1}\times \{1\}}=\mathrm{id}$. To complete the proof it suffices to define the homeomorphism $\Psi\colon \mathbb S^{n-1}\times [0,1] \to \mathbb S^{n-1}\times [0,1]$ by Proposition 3. Let $e_i, e'_i\colon \mathbb{B}^n\to \operatorname{int}\mathbb{B}^n$, $i\in \{1,\dots,k\}$, be orientation-preserving topological embeddings such that 1) the spheres $e_{i}(\partial \mathbb{B}^n)$ and $ e'_i(\partial \mathbb{B}^n)$ are locally flat in $\mathbb{B}^n$ for all $i\in \{1,\dots,k\}$; 2) $e_i(\mathbb{B}^n)\cap e_j(\mathbb{B}^n)=\varnothing$ and $e'_i(\mathbb{B}^n)\cap e'_j(\mathbb{B}^n)=\varnothing$ for all $i,j\in \{1,\dots,k\}$, $i\neq j$. Then there exists a homeomorphism $h\colon \mathbb{B}^n\to \mathbb{B}^n$ such that: 1) $h|_{\partial\mathbb{B}^n}=\mathrm{id}$; 2) $he_i=e_i'$, $i\in \{1,\dots,k\}$. Proof. In the case of smooth embeddings $e_i$ 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 $K_1=\mathbb{B}^n\setminus e_1(\operatorname{int} \mathbb{B}^n)$ and $K'_1=\mathbb{B}^n\setminus e'_1(\operatorname{int} \mathbb{B}^n)$ are homeomorphic to the standard annulus $\mathbb{S}^{n-1}\times [0,1]$. Let $\varphi_1\colon \mathbb{S}^{n-1}\times [0,1]\to K_1$ and $\varphi'_1\colon \mathbb{S}^{n-1}\times [0,1]\to K'_1$ be orientation-preserving homeomorphisms such that $\varphi_1(\mathbb{S}^{n-1}\times \{0\})=\varphi'_1(\mathbb{S}^{n-1}\times \{0\})=\partial \mathbb{B}^n.$ Let the maps $\psi_1\colon \mathbb{S}^{n-1}\times \{0\}\to \mathbb{S}^{n-1}\times \{0\}$ and $\eta_1\colon \mathbb{S}^{n-1}\times \{1\}\to \mathbb{S}^{n-1}\times \{1\}$ be defined by
$$
\begin{equation*}
\psi_1(x)=\varphi'_1\varphi_1^{-1}|_{\mathbb{S}^{n-1}\times \{0\}}\quad\text{and}\quad \eta_1(x)=\varphi'_1 e'_1 e^{-1}_1\varphi_1^{-1}|_{\mathbb{S}^{n-1}\times \{1\}},
\end{equation*}
\notag
$$
respectively. By definition, both $\psi_1$ and $\eta_1$ preserve orientation, and therefore there exist isotopies $\psi_{1,t}\colon \mathbb{S}^{n-1}\to \mathbb{S}^{n-1}$ and $\eta_{1,t}\colon \mathbb{S}^{n-1}\to \mathbb{S}^{n-1}$, $t\in [0,1]$, joining them with the identity map. Let $\varepsilon\in (0,1/3)$. Let the homeomorphisms $ G_1\colon \mathbb{S}^{n-1}\times [0,1]\to \mathbb{S}^{n-1}\times [0,1]$ and $h_1\colon \mathbb{B}^n\to \mathbb{B}^n$ be defined by
$$
\begin{equation*}
G_1(x,t)=\begin{cases} \biggl(\psi_{1,t/\varepsilon}(x), \dfrac{t}{\varepsilon}\biggr), &t\in [0,\varepsilon], \\ (x,t), &t\in [\varepsilon, 1-\varepsilon], \\ \biggl(\eta_{1,(1- t)/\varepsilon}(x), \dfrac{1- t}{\varepsilon}\biggr), &t\in [1-\varepsilon,1], \end{cases}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
h_1(x)=\begin{cases} {\varphi'_1}^{-1}(G_1(\varphi_1(x))),& x\in K_1, \\ x, &x\in e_1(\mathbb{B}^n), \end{cases}
\end{equation*}
\notag
$$
respectively.
It is easily checked that for $k=1$ the homeomorphism $h_1$ is the required one. Now assume that we have already constructed a homeomorphism $h_j\colon \mathbb{B}^n\to \mathbb{B}^n$ such that: 1) $h_j|_{\partial \mathbb{B}^n}=\mathrm{id}$; 2) $h_je_i=e_i'$, $1\leqslant i\leqslant j$. Let us construct a homeomorphism $h_{j+1}\colon \mathbb{B}^n\to \mathbb{B}^n$ satisfying 1) $h_{j+1}|_{\partial \mathbb{B}^n}=\mathrm{id}$; 2) $h_{j+1}e_i=e_i'$, $1\leqslant i\leqslant j+1$. We set $K_{j+1}=\mathbb{B}^n\setminus e_{j+1}(\operatorname{int} \mathbb{B}^n)$ and $K'_{j+1}=\mathbb{B}^n\setminus e'_{j+1}(\operatorname{int} \mathbb{B}^n)$ and denote by $\varphi_{j+1}\colon \mathbb{S}^{n-1}\times [0,1]\to K_{j+1}$ and $\varphi'_{j+1}\colon \mathbb{S}^{n-1}\times [0,1]\to K'_{j+1}$ orientation-preserving homeomorphisms such that $\varphi_{j+1}(\mathbb{S}^{n-1}\times \{0\})=\varphi'_{j+1}(\mathbb{S}^{n-1}\times \{0\})=\partial \mathbb{B}^n$. Let the maps $\psi_{j+1}\colon \mathbb{S}^{n-1}\times \{0\}$, $\eta_{j+1}\colon \mathbb{S}^{n-1}\times \{1\}$ be defined by
$$
\begin{equation*}
\psi_{j+1}(x)=\varphi'_{j+1}\varphi_{j+1}^{-1}|_{\mathbb{S}^{n-1}\times \{0\}}\quad\text{and} \quad \eta_{j+1}(x)=\varphi'_{j+1} e'_{j+1} e^{-1}_{j+1}\varphi_{j+1}^{-1}|_{\mathbb{S}^{n-1}\times \{1\}},
\end{equation*}
\notag
$$
respectively. By definition $\psi$ and $ \eta$ preserve orientation, and so there exit isotopies $\psi_{j+1,t}\colon \mathbb{S}^{n-1}\to \mathbb{S}^{n-1}$ and $\eta_{j+1,t}\colon \mathbb{S}^{n-1}\to \mathbb{S}^{n-1}$, $t\in [0,1]$, joining them with the identity map. Let $\varepsilon\in (0,1/3)$ be such that the set $C=\{(x,t)\mid x\in \mathbb{S}^{n-1}$, $t\in [0,\varepsilon]\cup [1-\varepsilon, 1]\}$ is disjoint from the balls $\varphi_{j+1}(e_i(\mathbb{B}^n))$ and $ \varphi'_{j+1}(e'_i(\mathbb{B}^n))$ for all $i\in \{1,\dots,j\}$. Let the homeomorphisms $ G_{j+1}\colon \mathbb{S}^{n-1}\times [0,1]\to \mathbb{S}^{n-1}\times [0,1]$ and $h_{j+1}\colon \mathbb{B}^n\to \mathbb{B}^n$ be defined by
$$
\begin{equation*}
G_{j+1}(x,t)= \begin{cases} \biggl(\psi_{j+1,t/\varepsilon}(x), \dfrac{t}{\varepsilon}\biggr), &t\in [0,\varepsilon], \\ (x,t), &t\in [\varepsilon, 1-\varepsilon], \\ \biggl(\eta_{j+1,(1- t)/\varepsilon}(x), \dfrac{1- t}{\varepsilon}\biggr), &t\in [1-\varepsilon,1], \end{cases}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
h_{j+1}(x)=\begin{cases} {\varphi'_{j+1}}^{-1}(G_{j+1}(\varphi_{j+1}(x))), &x\in K_{j+1}, \\ x, &x\in e_{j+1}(\mathbb{B}^n), \end{cases}
\end{equation*}
\notag
$$
respectively.
A direct verification shows that $h_{j+1}$ is the required homeomorphism. Proposition 3 is proved. Let $e_i\colon \mathbb{B}^{n}\to \operatorname{int} \mathbb{B}^n$, $i\in \{1,\dots,k\}$, be topological embeddings satisfying the assumptions of Proposition 3. A manifold with boundary that is homeomorphic to the manifold $\mathbb{B}^n\setminus \bigcup _{i=1}^k e_i(\operatorname{int}\mathbb{B}^n)$ 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 $H^n_k=\mathbb{B}^k\times \mathbb{B}^{n-k}$. We say that an $n$-manifold $M$ is obtained from an $n$-manifold $N$ with boundary $\partial N$ by attaching a handle $H^n_k$ if there exists an embedding $\psi$: $\partial {\mathbb B}^k\times \mathbb{B}^{n-k} \to \partial N$ such that $M=N\bigcup_{\psi} H^n_k$. A manifold $Q^n_k$ obtained from the ball $\mathbb{B}^n$ by attaching handles of index $\leqslant k$ 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 $Q^n_1$ is the genus of this body. 2.3. Some facts from graph theoryA graph $\Gamma$ is a union of two sets: a finite set $V(\Gamma)$ of elements called vertices and a finite set $E(\Gamma)$ of elements called edges. The edge set of a graph is formed by some pairs of its (not necessarily distinct) vertices. Two vertices $v, w\in V(\Gamma)$ are adjacent if the pair $v$, $w$ belongs to the set $E(\Gamma)$; 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 $\Gamma$ 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 $v_0=v,v_1,\dots,v_k=w$, 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 $\Gamma$ 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 $\Gamma$ be a graph with $r$ edges and $b$ vertices. Then the following assertions are equivalent: 1) $\Gamma$ is a tree; 2) $\Gamma$ is a connected graph and $b=r+1$; 3) $\Gamma$ is an acyclic graph and $b=r+1$; 4) any two different vertices are connected by a unique simple route; 5) $\Gamma$ 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 $\Gamma$, the rank of a vertex is defined as follows. We associate with $\Gamma$ a sequence of trees $\Gamma_0=\Gamma, \Gamma_1,\dots, \Gamma_r$ such that, for any $i\in \{1,\dots, r\}$, the tree $\Gamma_i$ is obtained from $\Gamma_{i-1}$ by removing all leaves and all edges incident to these leaves, and the tree $\Gamma_r$ consists either of one vertex or of two vertices joined by an edge. In the first case $\Gamma$ is called a central tree, and in the second case, a bicentral tree. The vertices and edge (if exists) of the tree $\Gamma_r$ are called the central vertices and edge of $\Gamma$. The rank of a vertex $v\in V(\Gamma)$ is defined by
$$
\begin{equation*}
\operatorname{rank}(v)=\max\{i\mid v\in V(\Gamma_{i}), \, i\in \{0,\dots,r\}\}.
\end{equation*}
\notag
$$
This definition shows that if an edge $(v,w)$ is not central, then ${|{\operatorname{rank}(v)\!-\!\operatorname{rank}(w)}|\!=\!1}$, 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 $\Gamma$ and $\Gamma'$ are called isomorphic if there exists a bijection $\xi\colon V(\Gamma)\to V(\Gamma')$ (called an isomorphism) such that if two vertices $v, w\in V(\Gamma)$ are adjacent, then so are also the vertices $\xi(u),\xi(v)\in V(\Gamma')$. § 3. Necessary and sufficient conditions for the topological equivalence of $G(\mathcal{S}^n_g)$-flows3.1. The scheme of a flow as a topological invariantLet $\varphi\colon M^n\to\mathbb{R}$ be a $C^r$-smooth function on a manifold $M^n$, $r\geqslant 2$. A point $p\in M^n$ is called a critical point of $\varphi$ if
$$
\begin{equation*}
\frac{\partial{\varphi}}{\partial{x}_1}(p)=\dots=\frac{\partial{\varphi}}{\partial{x}_n}(p)=0
\end{equation*}
\notag
$$
in local coordinates in a neighbourhood of the point $p$. A point $q\in M^n$ that is not critical is called a regular point of $\varphi$. A critical point $p$ of $\varphi$ is called nondegenerate if the Hessian matrix $\biggl(\dfrac{\partial^2{\varphi}}{\partial{x}_i\, \partial{x}_j}\biggr)\bigg|_p$ is nonsingular. A function $\varphi\colon M^n\to\mathbb R$ 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. Assertion 8. Let $f^t\in G(\mathcal{S}^n_g)$. Then there exists a Morse function $\varphi \colon M^n\to [0,n]$ such that: Set Let $\sigma_1\in \Omega^1_{f^t}$, $\sigma_{n-1}\in \Omega^{n-1}_{f^t}$ be arbitrary points. According to Theorem 2.3 in [13], there exist points $\omega\in \Omega^0_{f^t}$ and $\alpha\in \Omega^n_{f^t}$ such that $\operatorname{cl} W^{\mathrm s}_{\sigma_1}=W^{\mathrm s}_{\sigma_1}\cup \alpha$ and $\operatorname{cl} W^{\mathrm u}_{\sigma_{n-1}}=W^{\mathrm u}_{\sigma_{n-1}}\cup \omega$. Since the function $\varphi$ decreases along the trajectories of the flow $f^t$, the sets $W^{\mathrm s}_{\sigma_1}\cap \Sigma_{f^t}$ and $W^{\mathrm u}_{\sigma_{n-1}}\cap \Sigma_{f^t}$ are nonempty. We set
$$
\begin{equation*}
\mathcal{C}^{\mathrm s}_{f^t}=\bigcup_{\sigma_1\in \Omega^1_{f^t}}(W^{\mathrm s}_{\sigma_1}\cap \Sigma_{f^t})\quad\text{and} \quad \mathcal{C}^{\mathrm u}_{f^t}=\bigcup_{\sigma_{n-1}\in \Omega^{n-1}_{f^t}}(W^{\mathrm u}_{\sigma_{n-1}}\cap \Sigma_{f^t}).
\end{equation*}
\notag
$$
Definition 3. The triple $S_{f^t}=\{\Sigma_{f^t}, \mathcal{C}^{\mathrm s}_{f^t}, \mathcal{C}^{\mathrm u}_{f^t}\}$ is called the scheme of the flow $f^t\in G(\mathcal{S}^n_g)$, and the set $\Sigma_{f^t}$ is called its characteristic section. In Lemma 1 below we show that the topology of the section $\Sigma_{f^t}$ and the mutual arrangement of the connected components of the set $\mathcal{C}^{\mathrm s}_{f^t}\cup \mathcal{C}^{\mathrm u}_{f^t}$ on $\Sigma_{f^t}$ determine fully the class of topological equivalence of the flow $f^t\in G(\mathcal{S}^n_g)$, 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 $S_{f^t}=\{\Sigma_{f^t}, \mathcal{C}^{\mathrm s}_{f^t}, \mathcal{C}^{\mathrm u}_{f^t}\}$ and $S_{{f'}^t}=\{\Sigma'_{{f'}^t}, \mathcal{C}^{\mathrm s}_{{f'}^t}, \mathcal{C}^{\mathrm u}_{{f'}^t}\}$ are called equivalent if there exists a homeomorphism $h\colon \Sigma_{f^t}\to \Sigma'_{{f'}^t}$ such that Lemma 1. Two flows $f^t, {f'}^t\in G(\mathcal{S}^n_g)$ are topologically equivalent if and only if so are their schemes $S_{f^t}$ and $S_{{f'}^t}$. Proof. Set $A_{f^t}=\Omega^0_{f^t}\cup W^{\mathrm u}_{\Omega^1_{f^t}}$, $R_{f^t}=\Omega^n_{f^t}\cup W^{\mathrm s}_{\Omega^{n-1}_{f^t}}$ and $V_{f^t}=\mathcal{S}^n_g\setminus (A_{f^t}\cup R_{f^t})$ and denote the analogous objects for the flow ${f'}^t$ by $A_{{f'}^t}$, $R_{{f'}^t}$ and $V_{{f'}^t}$.
Let us prove the necessity in the lemma. Assume that two flows $f^t, {f'}^t\in G(\mathcal{S}^n_g)$ are topologically equivalent via a homeomorphism $\widetilde h$. Since $\Sigma'_{{f'}^t}$ is a level surface of the energy function of the flow, which is strictly decreasing on $V_{{f'}^t}$, for each point $y\in V_{{f'}^t}$ there exists a unique point $t_y\in \mathbb{R}$ such that ${f'}^{t_y}(x)\subset \Sigma'_{{f'}^t}$. We define the homeomorphism $h\colon \Sigma_{f^t}\to \Sigma'_{{{f'}^t}}$ by
$$
\begin{equation*}
h(x)={f'}^{t_{\widetilde h(x)}}(\widetilde h(x)), \qquad x\in \Sigma_{f^t}.
\end{equation*}
\notag
$$
Let $x$ lie in the manifold $W^{\mathrm s}_{\sigma_1}$ of some saddle point $\sigma_1\in \Omega^1_{f^t}$. Then $\widetilde h(\sigma_1)\in \Omega^1_{{f'}^t}$, $\widetilde h(W^{\mathrm s}_{\sigma})=W^{\mathrm s}_{h(\sigma_1)}$ and $h(W^{\mathrm s}_{\sigma_1}\cap \Sigma_{f^t})=W^{\mathrm s}_{\widetilde h(\sigma_1)}\cap \Sigma'_{{{f'}^t}}$. A similar analysis shows that $h(W^{\mathrm u}_{\sigma_{n-1}}\cap \Sigma_{f^t})=W^{\mathrm u}_{\widetilde h(\sigma_{n-1})}\cap \Sigma'_{{{f'}^t}}$ for any point $\sigma_{n-1}\in \Omega^{n-1}_{f^t}$. So the schemes $S_{f^t}$ and $ S_{{f'}^t}$ are equivalent. Let us prove sufficiency in the lemma. Assume that the schemes $S_{f^t}$ and $S_{{f'}^t}$ are equivalent and $h\colon \Sigma_{f^t}\to \Sigma'_{{{f'}^t}}$ is a homeomorphism such that $h(\mathcal{C}^{\mathrm s}_{f^t})=\mathcal{C}^{\mathrm s}_{{f'}^t}$ and $h(\mathcal{C}^{\mathrm u}_{f^t})=\mathcal{C}^{\mathrm u}_{{f'}^t}$. For any point $x\in \Sigma_{f^t}$ we set $x'=h(x)$ and denote by $\mathcal O_x$ and $\mathcal O'_{x'}$ the trajectories of the flows $f^t$ and ${f'}^t$ passing through the points $x$ and $x'$, respectively. Let $\varphi, \varphi'\colon \mathcal{S}^n_g\to [0,n]$ be the self-indexing energy functions of $f^t$ and ${f'}^t$, respectively. For any point $y\in V_{f^t}$ there exist a unique point $x\in \Sigma_{f^t}$ and a unique number $c_y\in (0, n)$ such that $y=\mathcal O_x\cap \varphi^{-1}(c_y)$. Let the homeomorphism $H\colon V_{f^t}\to V_{{f'}^t}$ be defined by The homeomorphism $h$ maps the set $\mathcal{C}^{\mathrm s}_{f^t}$ $(\mathcal{C}^{\mathrm u}_{f^t})$ to the set $\mathcal{C}^{\mathrm s}_{{f'}^t}$ (to $\mathcal{C}^{\mathrm u}_{{f'}^t}$, respectively) and so the homeomorphism $H$ thus constructed sends all stable ( unstable) separatrices of dimension $n-1$ of saddle equilibria of the flow $f^t$ 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 $c\in (0,1)$,
$$
\begin{equation*}
H\biggl(\varphi^{-1}(c)\setminus \bigcup_{\sigma\in \Omega^1_{f^t}} W^{\mathrm u}_{\sigma}\biggr)={\varphi'}^{-1}(c)\setminus \bigcup_{\sigma'\in \Omega^1_{{f'}^t}}W^{\mathrm u}_{\sigma'}
\end{equation*}
\notag
$$
and $\bigcup_{\sigma\in \Omega^1_{f^t}}W^{\mathrm u}_{\sigma}\cap\varphi^{-1}(c)$ and $\bigcup_{\sigma'\in \Omega^1_{{f'}^t}}W^{\mathrm u}_{\sigma'}\cap {\varphi'}^{-1}(c)$ are sets of the same cardinality. Hence $H$ extends to $\bigcup_{\sigma\in \Omega^1_{f^t}}W^{\mathrm u}_{\sigma}$ by continuity. In a similar way $H$ extends to $\bigcup_{\sigma\in \Omega^{n-1}_{f^t}}W^{\mathrm s}_{\sigma}$ and further to $\Omega^0_{f^t}\cup \Omega^{n}_{f^t}$.
Lemma 1 is proved. 3.2. The topology of the characteristic section $\Sigma_{f^t}$ and of connected components of $\mathcal{C}^{\mathrm s}_{f^t}\cup \mathcal{C}^{\mathrm u}_{f^t}$We recall that, given a flow $f^t\in G(\mathcal{S}^n_g)$, we denote by $\nu_{f^t}$ and $\mu_{f^t}$ the numbers of nodal and saddle equilibria, respectively; the quantity $g_{f^t}$ is defined by By Assertion 1 we have $g=g_{f^t}$. Lemma 2. Let $f^t\in G(\mathcal{S}^n_g)$. Then the following hold. 1. The characteristic section $\Sigma_{f^t}$ is the boundary of an $(n,1)$-handlebody of genus $g$. 2. For all $\sigma_1\subset \Omega^1_{f^t}$ and $\sigma_{n-1}\subset \Omega^{n-1}_{f^t}$ each connected component $l^{\mathrm s}_{\sigma_1}=W^{\mathrm s}_{\sigma_1} \cap \Sigma_{f^t}$ and $l^{\mathrm u}_{\sigma_{n-1}}=W^{\mathrm u}_{\sigma_{n-1}} \cap \Sigma_{f^t}$ of the set $\mathcal{C}^{\mathrm s}_{f^t}$ or $\mathcal{C}^{\mathrm u}_{f^t}$, respectively, is a sphere of dimension $n-2$ which is smoothly embedded in $\Sigma_{f^t}$. 3. If $g>0$, then there exist two sets of saddle equilibrium states $\sigma_1^1, \dots,\sigma_{1}^{g}\in \Omega^1_{f^t}$ and $\sigma_{n-1}^1, \dots,\sigma_{n-1}^{g}\in \Omega^{n-1}_{f^t}$ and sets $\{T^{\mathrm s}_i\subset \Sigma_{f^t}\}$ and $\{T^{\mathrm u}_j \subset \Sigma_{f^t}\}$ of pairwise disjoint tubular neighbourhoods of the spheres $\{l^{\mathrm s}_{\sigma_1^i}\}$ and $\{l^{\mathrm u}_{\sigma_{n-1}^j}\}$, respectively, such that the sets $\operatorname{cl} (\Sigma_{f^t} \setminus \bigcup _{i=1}^{g} T^{\mathrm s}_i)$ and $\operatorname{cl} (\Sigma_{f^t} \setminus \bigcup _{j=1}^{g} T^{\mathrm u}_j)$ are homeomorphic to the sphere with $2{g}$ holes. Proof. Let us prove part 1. We set $Q_a=\varphi^{-1}([0,n/2])$, where $\varphi$ is the self-indexing energy function of the flow $f^t$. By definition $\Sigma_{f^t}=\partial Q_a$. First we prove that $Q_a$ is connected.
Since $\varphi$ decreases along the trajectories of $f^t$, we have $Q_a\subset \bigcup_{p\in \Omega^0_{f^t}\cup \Omega^1_{f^t}} W^{\mathrm s}_{p}$. We set $U_a=\bigcup_{i\in \mathbb{Z}}f^i(Q_a)$ and $R_{f^t}=\Omega^n_{f^t}\cup W^{\mathrm s}_{\Omega^{n-1}_{f^t}}$. According to Theorem 2.3 in [13], the manifold $\mathcal{S}^{n}_g$ can be represented as $\mathcal{S}^{n}_g=\bigcup_{p\in \Omega^0_{f^t}\cup \Omega^1_{f^t}}{W^{\mathrm s}_p}\cup R_{f^t}$. Hence $U_a=\mathcal{S}^{n}_g\setminus R_{f^t}$. Since $n\geqslant 3$ and the dimension of the set $R_{f^t}$ is at most 1, it follows that $U_a$ is connected (see [32], Ch. 4, Theorem 4). Hence $Q_a$ is too. Indeed, if $Q_a$ were disconnected, then it could be represented as the union of nonempty disjoint closed subsets $E_1$ and $E_2$ such that if $x\in E_i$, then $f^t(x)\in E_i$ for any $t\geqslant 0$. Hence the set $U_a=\bigcup_{t\in \mathbb{R}} (f^t(E_1)\cup f^t(E_2))$ would also be disconnected (if the intersection $\bigcup_{t\in \mathbb{R}} f^t(E_1)\cap \bigcup_{t\in \mathbb{R}} f^t(E_2)$ is nonempty, then for any point $x$ in this intersection there exists $t_x$ such that $f^{t_x}(x)\in E_1\cap E_2$, but this contradicts the assumption $E_1\cap E_2=\varnothing$). It follows from Morse’s lemma that for any $\varepsilon\,{\in}\, (0,1)$ the manifold ${M_\varepsilon\!=\!\varphi^{-1}([0,\varepsilon])}$ is homeomorphic to the disjoint union $k^0=|\Omega_{f^t}^0|$ 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 $Q_a=\varphi^{-1}([0,n/2])$ is obtained from $M_\varepsilon$ by attaching $k^1=|\Omega_{f^t}^1|$ handles $H_1,\dots,H_{k^1}$ 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 $f^t$, the section $\Sigma_{f^t}$ and the handlebody $Q_a$ in the case when $g=0$ and $k_1=1$). Since $Q_a$ is connected, we have $k^1\geqslant k^0-1$. We set $g_a=k^1-k^0+1$. We claim that $Q_a$ is an $(n,1)$-handlebody of genus $g_a$. There are two cases to consider: $g_a=0$ and $g_a>0$. Let $g_a=0$. We use induction on $k^0$ to show that the compact connected manifold obtained from $k^0$ copies of closed $n$-balls by attaching $(k^0-1)$ handles of index $1$ is homeomorphic to the $n$-ball. This will imply that, in the case $g_a=0$, $Q_a$ is homeomorphic to an $n$-ball (that is, to a handlebody of genus $0$). For $k^0=1$ there is one ball and no handles, and in this case the claim is true. Assuming that the claim holds for $k^0=i\geqslant 1$ consider the case $k^0=i+1$. In this case $Q_a$ 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 $g_{a}>0$. Then $Q_a$ is connected and obtained by attaching $k^1=k^0-1+g_a$ handles to $k^0$ balls. According to the above, the connected manifold obtained by attaching $k^0-1$ handles to $k_0$ balls is homeomorphic to the ball. Attaching $g_a$ more handles to this manifold we obtain an $(n,1)$-handlebody of genus $g_a$. So $Q_a$ is a handlebody of genus $g_a= k^1-k^{0}+1$. In a similar way, considering the flow $f^{-t}$ and its energy function $\psi=n-\varphi$, we find that the manifold $Q_r=M^n\setminus \operatorname{int} Q_a$ is an $(n,1)$-handlebody of genus $g_r=k^{n-1}-k^n+1$, where $k^n=|\Omega^n_{f^t}|$, $k^{n-1}=|\Omega^{n-1}_{f^t}|$. The manifolds $Q_a$ and $Q_r$ have the common boundary, and so $g_a=g_r$. By the definition of the quantities $g_{f^t}$, $g_a$ and $g_r$,
$$
\begin{equation*}
\frac{g_a+g_r}2=g_{a}=\frac{k^{n-1}+k^{1}-k^0-k^{n}+2}2 =\frac{\mu_{f^t}-\nu_{f^t}+2}2=g_{f^t}.
\end{equation*}
\notag
$$
An appeal to Assertion 1 shows that $g_{f^t}=g$. Thus, $g_a=g$. Let us prove part 2. Assertion 8 shows that for any point $\sigma_1\in \Omega^1_{f^t}$ the intersection of $W^{\mathrm s}_{\sigma_1}$ and $\Sigma_{f^t}$ is transversal; hence the set $l^{\mathrm s}_{\sigma_1}=W^{\mathrm s}_{\sigma_1} \cap \Sigma_{f^t}$ is a smooth closed manifold of dimension $n - 1 + n - 1 - n=n-2$. Since $\Sigma_{f^t}$ is a section for the trajectories of $f^t$, the submanidold $l^{\mathrm s}_{\sigma_1}$ is a section for all trajectories of the restriction of the flow $f^t$ to the set $W^{\mathrm s}_{\sigma_1}\setminus \sigma_1$. Consider the retraction
$$
\begin{equation*}
r\colon W^{\mathrm s}_{\sigma_1}\setminus {\sigma_1}\to l^{\mathrm s}_{{\sigma_1} }
\end{equation*}
\notag
$$
that associates with a point $x\in W^{\mathrm s}_{\sigma_1}\setminus {\sigma_1}$ the point $y=f^{t_x}(x)\in l^{\mathrm s}_{{\sigma_1}}$ of intersection of the set $l^{\mathrm s}_{{\sigma_1}}$ and the trajectory of the flow $f^t$ passing through $x$. The retraction $r$ is connected with the identity map $\mathrm{id}\colon W^{\mathrm s}_{\sigma_1}\setminus {\sigma_1}\to W^{\mathrm s}_{\sigma_1}\setminus {\sigma_1}$ via the homothety Therefore, $l^{\mathrm s}_{{\sigma_1}}$ has the same homotopy type as $W^{\mathrm s}_{\sigma_1}\setminus {\sigma_1}$. The manifold $W^{\mathrm s}_{{\sigma_1}}\setminus {\sigma_1}$ is homeomorphic to $\mathbb{R}^{n-1}\setminus O$, so it has the homotopy type of the $(n-2)$-sphere. Now, by Poincaré’s theorem the manifold $l^{\mathrm s}_{{\sigma_1}}$ is homeomorphic to the $(n-2)$-sphere. The arguments for a point $\sigma_{n-1}\in \Omega^{n-1}_{f^t}$ are similar. Let us prove assertion 3 of Lemma 2. Let $g>0$. Since $Q_a$ is a handlebody of genus $g$, there exist smooth embeddings $e_i$: $[-1,1]\times \mathbb{B}^{n-1}\to Q_a$, $i\in \{1,\dots, g\}$, such that:
We set $H_i=e_i([-1,1]\times \mathbb{B}^{n-1})$, $B_i^-=e_i(\{-1\}\times \mathbb{B}^{n-1})$ and $B_i^+=e_i(\{1\}\times \mathbb{B}^{n-1})$, $i\in \{1, \dots, g\}$. By the construction of $Q_a$ (see the proof of part 1 of Lemma 2) each subset $H_i$ has a nonempty intersection with the stable manifold of at least one saddle equilibrium $\sigma_1^i\in \Omega^1_{f^t}$. By Assertion 3 the sphere $l^{\mathrm s}_{\sigma_1^i}$ bounds a ball $B^{\mathrm s}_{\sigma_1^i}\in W^{\mathrm s}_{\sigma_1^i}$, which contains the point $\sigma_1^i$. Therefore, $\operatorname{int} B^{\mathrm s}_{\sigma^i_1}\subset \operatorname{int} H_i$ and $B^{\mathrm s}_{\sigma_1^i}=W^{\mathrm s}_{\sigma_1^i}\cap H_i$. The balls $\{B^{\mathrm s}_{\sigma_1^i}\}$ are smoothly embedded in $Q_a$, and so, for each $i\in \{1,\dots, g\}$ there exists a smooth embedding $\psi_i\colon [-1,1]\times \mathbb{B}^{n-1}\to H_i$ with the following properties:
The disc $B^{\mathrm s}_{\sigma_1^i}$ partitions $H_i$ into two connected components, and therefore $\operatorname{cl} (H_i\setminus N_i)$ 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 $B_i^+\cup \widetilde{B}_i^+\subset H_i^+$. By Assertion 4 the set $\partial H_i^+\setminus (\operatorname{int} B_i^+\cup \operatorname{int} \widetilde{B}_i^+)$ (which also lies in the boundary of $H_i$) is homeomorphic to the annulus $\mathbb{S}^{n-2}\times [0,1]$. Now it follows from Corollary 3 that $\partial H_i^+$ is homeomorphic to the sphere $S^{n-1}$, and an appeal to Assertion 2 shows that $H_i^+$ is homeomorphic to the ball $B^n$. A similar analysis shows that $H_i^-$ is homeomorphic to the ball $B^n$. By definition $\operatorname{cl} (Q_a\setminus \bigcup_{i=1}^{g}H_{i})$ is homeomorphic to the $n$-ball. Hence the set
$$
\begin{equation*}
P_a=\operatorname{cl} \biggl(Q_a\setminus \bigcup_{i=1}^{g}N_{i}\biggr) =\operatorname{cl} \biggl(Q_a\setminus \bigcup_{i=1}^{g}H_{i}\biggr) \cup \bigcup_{i=1}^{g}(H^-_i\cup H^+_i)
\end{equation*}
\notag
$$
is also homeomorphic to the $n$-ball. Setting $T^{\mathrm s}_i=N_i\cap \Sigma_{f^t}$, we have
$$
\begin{equation*}
\operatorname{cl} \biggl(\Sigma_{f^t}\setminus \bigcup_{i=1}^{g} T^{\mathrm s}_i\biggr) =\partial P_a\setminus \biggl(\bigcup_{i=1}^{g}\operatorname{int}\widetilde{B}_i^-\cup \operatorname{int}\widetilde{B}^+_i\biggr);
\end{equation*}
\notag
$$
therefore, $\operatorname{cl} (\Sigma_{f^t}\setminus \bigcup_{i=1}^{g} T^{\mathrm s}_i)$ 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 $\sigma_{n-1}^1, \dots,\sigma_{n-1}^{g}\in \Omega^{n-1}_{f^t}$ exists since the above arguments also apply to the flow $f^{-t}$ and its energy function $n-\varphi$. The following result is a direct consequence of part 3 of Lemma 2. Corollary 5. If $g>0$, then each of the sets $C^{\mathrm s}_{f^t}$ and $C^{\mathrm u}_{f^t}$ contains at least one set $\{l^{\mathrm s}_{\sigma_1^1},\dots, l^{\mathrm s}_{\sigma_1^g}\}$ or $\{l^{\mathrm u}_{\sigma_{n-1}^1}, \dots, l^{\mathrm u}_{\sigma_{n-1}^g}\}$ such that for any $k\in [1, g]$ the manifolds $ \bigl(\Sigma_{f^t}\setminus \bigcup _{i=1}^k l^{\mathrm s}_{\sigma_1^i}\bigr)$ and $\bigl(\Sigma_{f^t}\setminus \bigcup _{i=1}^k l^{\mathrm u}_{\sigma_{n-1}^i}\bigr)$ are connected. In addition, any $g+1$ spheres in the set $C^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t}$ divide $\Sigma_{f^t}$ into several connected components. Definition 5. The set of saddle equilibria $\{\sigma_1^1,\dots,\sigma_1^g\}$ ($\{\sigma_{n-1}^1,\dots,\sigma_{n-1}^g\}$) described in part 3 of Lemma 2 and the set of spheres $\{l^{\mathrm s}_{\sigma_1^i}\}$ ($\{l^{\mathrm u}_{\sigma_{n-1}^j}\}$, respectively) corresponding to the intersection of their stable (respectively, unstable) manifolds with the section $\Sigma_{f^t}$ are called a maximal nonseparating $\mathrm s$-set (respectively, $\mathrm u$-set). 3.3. The relationship between between a bi-colour graph and the scheme of the flow $f^t\in G(\mathcal{S}^n_g)$Let $\eta_{f^t}\colon V(\Gamma_{f^t})\cup E(\Gamma_{f^t})\to \mathcal{D}_{f^t}\cup \mathcal{L}_{f^t}$ be a bijection such that $\eta_{f^t}(V(\Gamma_{f^t}))=\mathcal{D}_{f^t}$, $\eta_{f^t}(E(\Gamma_{f^t}))=\mathcal{L}_{f^t}$, and an edge $e\in E(\Gamma_{f^t})$ is incident to vertices $v, w\in V(\Gamma_{f^t})$ if and only if the sphere $\eta_{f^t}(e)\in \mathcal{L}_{f^t}$ lies on the boundary of the domains $\eta_{f^t}(v), \eta_{f^t}(w)\in \mathcal{D}_{f^t}$. Let $d_{f^t}$ be the set of connected components of the set $\Sigma_{f^t}\setminus (C^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t})$. By Lemma 2 each sphere $L\in \mathcal{L}_{f^t}$ (and therefore each domain $D\in \mathcal{D}_{f^t}$) intersects the characteristic section $\Sigma_{f^t}$ in precisely one connected component. Hence the bijection $\eta_{f^t}$ induces a bijection $\eta_{*}\colon V(\Gamma_{f^t}) \cup E(\Gamma_{f^t})\to d_{f^t}\cup (C^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t})$ with the same properties. In what follows the graph $\Gamma_{f^t}$ is identified with a one-dimensional polyhedron embedded in the characteristic section $\Sigma_{f^t}$ as in the following statement (see Figure 8). Proposition 4. The graph $\Gamma_{f^t}$ embeds in the characteristic section $\Sigma_{f^t}$ so that to each vertex $v\in V(\Gamma_{f^t})$ there corresponds a point $\widetilde{v}$ in the domain $\eta_{*}(v)\in d_{f^t}$, and to an edge $e\in E(\Gamma_{f^t})$ joining vertices $v, w\in V(\Gamma_{f^t})$ there corresponds a smooth arc $\widetilde{e}\subset \Sigma_{f^t}$ joining the points $\widetilde{v}$ and $ \widetilde{w}$ and intersecting the sphere $l=\eta_*(e)\in C^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t}$ in one point. In what follows the point $\widetilde{v}$ and the arc $\widetilde{e}$ are called a vertex and an edge of the graph $\Gamma_{f^t}$, respectively; the symbol $\sim$ is suppressed in the notation. We say that an edge $e$ separates the graph $\Gamma_{f^t}$ if the number of connected components of the graph $\Gamma_{f^t}\setminus e$ is greater than that of $\Gamma_{f^t}$. Proposition 5. 1. The graph $\Gamma_{f^t}$ is connected. 2. An edge $e\in \Gamma_{f^t}$ separates the graph $\Gamma_{f^t}$ if and only if the sphere $\eta_{*}(e)=l$ separates the section $\Sigma_{f^t}$. Proof. Let us prove claim 1. Let $v$ and $w$ be vertices of the graph $\Gamma_{f^t}$. It follows from Lemma 2 that the section $\Sigma_{f^t}$ is connected. Hence the points $v$ and $w$ can be connected by a path $\gamma\colon [0,1]\to \Sigma_{f^t}$. These points lie in different connected components of the set $\Sigma_{f^t}\setminus (C^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t})$, and so the image $\gamma([0,1])$ of the path $\gamma$ intersects some spheres $l_1,\dots, l_k\in C^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t}$ and traverses the connected components $d_1,\dots, d_{k+1}$ of the set $\Sigma_{f^t}\setminus (C^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t})$. Assume that $\gamma(0)=v\in d_1$ and $\gamma(1)=w\in d_k$. Then there is a route $\mathcal{M}$ in $\Gamma_{f^t}$ composed of vertices $v_1=v, v_2,\dots, v_{k+1}=w$ and edges $e_1,\dots,e_k$ such that $v_i\in d_i$, and the edge $e_i$ intersects the sphere $l_i$, $i\in \{1,\dots,k\}$ (Figure 9). Hence the graph $\Gamma_{f^t}$ is connected.
Let us prove claim 2. Let the vertices $v$ and $w$ be incident to an edge $e$ such that $\Gamma_{f^t}\setminus e$ is disconnected and $l=\eta_*(e)$. We claim that the set $\Sigma_{f^t}\setminus l$ is disconnected. Assume on the contrary that $\Sigma_{f^t}\setminus l$ is connected. Proceeding as in the proof of claim 1 we can construct a route $\mathcal{M}$ in $\Gamma_{f^t}$ that connects the points $v$ and $w$ and does not contain the edge $e$. The existence of such a route means that the graph $\Gamma_{f^t}\setminus e$ is connected, which contradicts the assumption. Therefore, $\Sigma_{f^t}\setminus l$ is disconnected. Now assume that $\Sigma_{f^t}\setminus l$ is disconnected. We claim that the graph $\Gamma_{f^t}\setminus e$ is too. Assume for a contradiction that $\Gamma_{f^t}\setminus e$ is connected. Then there exists a route $\mathcal{M}$ in $\Gamma_{f^t}\setminus e$ that connects the vertices $v$ and $w$. The route $\mathcal{M}$ does not intersect the sphere $l$, and therefore $v$ and $w$ lie in the same connected component of $\Sigma_{f^t}\setminus l$. The sets $\Sigma_{f^t}\setminus l$ and $e\setminus l$ are disconnected, and so $\Gamma_{f^t}\setminus l$ is too, and the vertices $v$ and $w$ incident to the edge $e$ lie in different connected components of the set $\Gamma_{f^t}\setminus l$. This contradiction shows that the edge $e$ cannot lie in a cycle of the graph $\Gamma_{f^t}\setminus e$. Proposition 5 is proved. Lemma 3. Let $f^t\in G(\mathcal{S}^n_g)$. If $g=0$, then $\Gamma_{f^t}$ is a tree. If $g>0$, then $\Gamma_{f^t}$ 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 $\Gamma_{f^t}$ contains both an edge of colour $\mathrm s$ and an edge of colour $\mathrm u$, and these edges correspond to spheres $l^{\mathrm s}_{\sigma_1^i}\in C^{\mathrm s}_{f^t}$ and $l^{\mathrm u}_{\sigma_{n-1}^j}\in C^{\mathrm u}_{f^t}$ which lie in maximal nonseparating $\mathrm s$- and $\mathrm u$-sets, respectively. Proof. Let $g=0$. By Lemma 2 the characteristic section $\Sigma_{f^t}$ is a sphere of dimension $n-1$, and for any sphere $l\in \mathcal{L}_{f^t}$ the intersection $l\cap \Sigma_{f^t}$ is a sphere of dimension $n-2$. We recall that $\mu_{f^t}$, the number of saddles in the flow $f^t$, is equal to the number of spheres in the set $\mathcal{L}_{f^t}$. By Corollary 1 the set $\Sigma_{f^t}\setminus \bigcup _{l\in \mathcal{L}_{f^t}}l$ has precisely $\mu_{f^t}+1$ connected components. Therefore, the graph $\Gamma_{f^t}$ has $\mu_{f^t}+1$ vertices and $\mu_{f^t}$ edges. By Proposition 5, $\Gamma_{f^t}$ is connected. Now, by Assertion 7 the graph $\Gamma_{f^t}$ is a tree.
Let $g>0$. By part 3 of Lemma 2 there exists a maximal nonseparating $\mathrm s$-set of spheres $l^{\mathrm s}_{\sigma_1^1}, \dots, l^{\mathrm s}_{\sigma_1^g}\in C^{\mathrm s}_{f^t}$ such that the set $\widetilde\Sigma_{f^t}=\Sigma_{f^t}\setminus \bigcup _{i=1}^gl^{\mathrm s}_{\sigma_1^i}$ is connected and, for any sphere $l\in C^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t}\setminus \bigcup _{i=1}^gl^{\mathrm s}_{\sigma_1^i}$, the set $\widetilde \Sigma_{f^t}\setminus l$ is disconnected. Let $e^{\mathrm s}_1, \dots, e^{\mathrm s}_g$ and $ e$ be the edges of $\Gamma_{f^t}$ corresponding to the spheres $l^{\mathrm s}_{\sigma_1^1}, \dots, l^{\mathrm s}_{\sigma_1^g}$ and $l$, respectively. From Proposition 5 it follows that the graph $\widetilde\Gamma_{f^t}=\Gamma_{f^t}\setminus \bigcup _{i=1}^{g}e^{\mathrm s}_i$ is connected and $\widetilde \Gamma_{f^t}\setminus e$ is disconnected. Hence $\widetilde \Gamma_{f^t}$ is connected and contains no cycles, so that it is a tree. Now it follows from Assertion 7 that each edge $e^{\mathrm s}_1, \dots, e^{\mathrm s}_g$ lies on a simple cycle of the graph $\Gamma_{f^t}$. No two edges $e^{\mathrm s}_i$, $e^{\mathrm s}_j$, $i\neq j$ lie on the same cycle since otherwise $\Gamma_{f^t}\setminus (e^{\mathrm s}_i\cup e^{\mathrm s}_j)$ would have two connected components. Further, no edge lies in two cycles simultaneously, since otherwise $\widetilde{\Gamma}_{f^t}$ would be disconnected (Figure 10). So $\Gamma_{f^t}$ contains at least $g$ pairwise distinct simple cycles, each of which contains an edge $e^{\mathrm s}_i$ of colour $\mathrm s$. Since $\widetilde{\Gamma}_{f^t}$ is acyclic, $\Gamma_{f^t}$ contains precisely $g$ cycles. By part 3 of Lemma 2, in addition to the maximal nonseparating $\mathrm s$-set, there also exists a maximal nonseparating $\mathrm u$-set of spheres $l^{\mathrm u}_{\sigma_{n-1}^1}, \dots, l^{\mathrm u}_{\sigma_{n-1}^g}$ in the set $C^{\mathrm u}_{f^t}$. To each sphere $l^{\mathrm u}_{\sigma_{n-1}^i}$ in this set there corresponds an edge $e^{\mathrm u}_i$ of $\Gamma_{f^t}$ coloured with colour $\mathrm u$. Replacing $\mathrm s$ by $\mathrm u$ in the above analysis we find that each of the $g$ simple cycles of $\Gamma_{f^t}$, along with the edge $e^{\mathrm s}_i$ of colour $\mathrm s$, also contains an edge $e^{\mathrm u}_i$ of colour $\mathrm u$. 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 $\xi$ between two graphs $\Gamma_{f^t}$ and $\Gamma_{{f'}^t}$ implies the equivalence of the schemes $S_{f^t}$ and $S_{{f'}^t}$. With this proviso, Lemma 1 would imply that the flows $f^t$ and ${f'}^t$ are topologically equivalent. Let $\eta_{f^t}\colon V(\Gamma_{f^t})\cup E(\Gamma_{f^t})\to \mathcal{D}_{f^t}\cup \mathcal{L}_{f^t}$ be a bijection such that an edge $e\in E(\Gamma_{f^t})$ is incident to vertices $v, w\in V(\Gamma_{f^t})$ if and only if the sphere $\eta_{f^t}(e)\in \mathcal{L}_{f^t}$ lies on the boundary of the domains $\eta_{f^t}(v), \eta_{f^t}(w)\in \mathcal{D}_{f^t}$ (see § 3.3). We denote by $\eta_{{f'}^t}\colon V(\Gamma_{{f'}^t})\cup E(\Gamma_{{f'}^t})\to \mathcal{D}_{{f'}^t}\cup \mathcal{L}_{{f'}^t}$ a bijection with the same properties with regard to the flow ${f'}^t$. The isomorphism $\xi\colon V(\Gamma_{f^t})\to V(\Gamma_{{f'}^t})$ induces an isomorphism $\xi_*\colon \mathcal{D}_{f^t}\to \mathcal{D}_{{f'}^t}$ as follows: $\xi_*=\eta_{{f'}^t} \xi \eta_{f^t}^{-1}|_{\mathcal{D}_{f^t}}$. Since $\xi$ preserves adjacency and the colours of edges, the isomorphism $\xi_*$ extends naturally to an isomorphism between the sets $\mathcal{L}_{f^t}$ and $\mathcal{L}_{{f'}^t}$; this isomorphism will also be denoted by $\xi_*$. Let $\mu_{f^t}$ and $\nu_{f^t}$ ($\mu_{{f'}^t}$ and $\nu_{{f'}^t}$) be the numbers of saddle and nodal equilibria of the flow $f^t$ (of ${f'}^t$, respectively). Since $\mu_{f^t}=|\mathcal{L}_{f^t}|=|E(\Gamma_{f^t})|$, $\mu_{{f'}^t}=|\mathcal{L}_{{f'}^t}|=|E(\Gamma_{{f'}^t})|$ and the graphs $\Gamma_{f^t}$ and $\Gamma_{{f'}^t}$ are isomorphic, it follows that $\mu_{f^t}=\mu_{{f'}^t}$. We claim that $\nu_{f^t}=\nu_{{f'}^t}$. The nodal states of the flow $f^t$ can be partitioned into two subsets: points of the first lie on spheres in $\mathcal{L}_{f^t}$, and in this case, as the graphs $\Gamma_{f^t}$ and $\Gamma_{{f'}^t}$ are isomorphic, for $f^t$ and ${f'}^t$ these sets have the same cardinality. Points of the second subset lie in domains in the set $\mathcal{D}_{f^t}$, each domain $D\in \mathcal{D}_{f^t}$ containing at most one sink or source equilibrium. In addition, the sink (source) equilibrium $\omega$ ($\alpha$, respectively) lies in a domain $D\in \mathcal{D}_{f^t}$ 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 $\mathrm{s} $ (of colour $\mathrm{u}$). Hence, since the graphs $\Gamma_{f^t}$ and $\Gamma_{{f'}^t}$ are isomorphic, it follows that $\nu_{f^t}=\nu_{{f'}^t}$. The type of the ambient manifold of the flow $f^t$ is a function of the quantity $g_{f^t}=(\mu_{f^t}-\nu_{f^t}+2)/2$, and therefore $g_{f^t}=g_{{f'}^t}$. Next we set Let $\Sigma_{f^t}$ and $\Sigma_{{f'}^t}$ be the characteristic sections of the flows $f^t$ and ${f'}^t$. By definition, each sphere $L\in \mathcal{L}_{f^t}$ ($L'\in \mathcal{L}_{{f'}^t})$ and each domain $D\in \mathcal{D}_{f^t}$ ($D'\in \mathcal{D}_{{f'}^t}$) intersects $\Sigma_{f^t}$ (intersects $\Sigma_{{f'}^t}$, respectively) in one connected component. Hence the isomorphism $\xi_*$ induces in a natural way a bijection between the elements of $C^{\mathrm u}_{f^t}$ and $C^{\mathrm u}_{{f'}^t}$, between $C^{\mathrm s}_{{f}^t}$ and $C^{\mathrm s}_{{f'}^t}$, and also between the connected components of the sets $\Sigma_{{f^t}}\setminus (C^{\mathrm u}_{f^t}\cup C^{\mathrm s}_{f^t})$ and $\Sigma_{{f'}^t}\setminus (C^{\mathrm u}_{{f'}^t}\cup C^{\mathrm s}_{{f'}^t})$, which we also denote by $\xi_*$. By the definition of $\xi_*$ a connected component $d\subset \Sigma_{{f^t}}\setminus (C^{\mathrm u}_{f^t}\cup C^{\mathrm s}_{f^t})$ corresponds to a connected component $d'=\xi_*(d)\subset \Sigma_{{f'}^t}\setminus (C^{\mathrm u}_{{f'}^t}\cup C^{\mathrm s}_{{f'}^t})$ if and only if, for any connected component $l\subset C^{\mathrm u}_{f^t}\cup C^{\mathrm s}_{f^t}$ of the boundary of the domain $d$, there exists a connected component $l'=\xi_*(l)\subset C^{\mathrm u}_{{f'}^t}\cup C^{\mathrm s}_{{f'}^t}$ of the boundary of $d'$. Now we construct a homeomorphism $h\colon \Sigma_{f^t}\to \Sigma_{{f'}^t}$ such that $h(l)=l'$ for any sphere $l\subset C^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t}$. There are two cases to consider, $g=0$ and $g>0$. Case 1: $g=0$. By Lemma 3 the graphs $\Gamma_{f^t}$ and $\Gamma_{{f'}^t}$ are trees, and by Lemma 2 the characteristic sections $\Sigma_{f^t}$ and $\Sigma_{{f'}^t}$ are homeomorphic to spheres. Let $r$ be the maximum rank of the vertices of $\Gamma_{f^t}$ and $\Gamma_{{{f'}^t}}$. There are two cases to consider:
Case 1, a). Let $v_0$ be the central vertex of $\Gamma_{f^t}$. If $r=0$, then $\Gamma_{f^t}$ consists of the unique vertex $v_0$, and the nonwandering set of the flow $f^t$ does not contain saddle equilibria. In this case the nonwandering set of $f^t$ consists of precisely two saddle equilibria, a source and a sink, and all such flows are topologically equivalent. Let $r>0$. Then the vertex $v_0$ has degree $\delta_0\geqslant 2$ (if $\delta_0=1$, then $v_0$ is a leaf; such a vertex is central only in the case when $r=1$ and the graph $\Gamma_{f^t}$ is bicentral). Let $v_{0,1}, \dots, v_{0,\delta_0}$ be the vertices adjacent to $v_0$. We denote by $l_{0,i}$ the sphere in $C^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t}$ that corresponds to the edge $(v_0, v_{0,i})$. We set $\mathfrak{l}_0=\bigcup _{i=1}^{\delta_0}l_{0,i}$. Then the boundary of the domain $d\in \mathcal{D}_{f^t}\cap \Sigma_{{f}^t}$ corresponding to the vertex $v_0$ consists precisely of all spheres in $\mathfrak{l}_0$ (see Figure 11). We denote the subset consisting of the spheres in $C^{\mathrm s}_{{f'}^t}\cup C^{\mathrm u}_{{f'}^t}$ corresponding to edges incident to the central vertex $v'_0\in V(\Gamma'_{f^t})$ by $\mathfrak{l}'_0$. We assume that the spheres in $\mathfrak{l}'_0$ are labelled so that $l'_{0,i}=\xi_*(l_{0,i})$, $i\in \{1,\dots,\delta_0\}$. By Assertion 3 each sphere $l_{0,i}\in \mathfrak{l}_0$ partitions $\Sigma_{{f}^t}$ into two connected components, and the closure of each component is homeomorphic to the $(n-1)$-ball. Hence $\mathfrak{b}_0=\Sigma_{f^t}\setminus \operatorname{cl} d$ is a union of pairwise disjoint open balls bounded by spheres in $\mathfrak{l}_0$. We denote the complement to the set $\operatorname{cl}d'\in \mathcal{D}_{{f'}^t}\cap \Sigma_{{f'}^t}$ corresponding to the vertex $v'_0$ by $\mathfrak{b}'_0$. Since the graphs $\Gamma_{f^t}$ and $ \Gamma_{{f'}^t}$ are isomorphic, $\mathfrak{b}'_0$ consists of the same number of balls as $\mathfrak{b}_0$, and for each ball $b_{0,i}\in \mathfrak{b}_0$ bounded by a sphere $l_{0,i}\in \mathfrak{l}_0$ there exists a ball $b'_{0,i}\in \mathfrak{b}'_0$ bounded by a sphere $l'_{0,i}\in \mathfrak{l}'_0$. It follows from Corollary 4 and Assertion 6 that there exists a homeomorphism $h_0\colon \Sigma_{{f}^t}\to \Sigma_{{f'}^t}$ such that $h_0(l_{0,i})=l'_{0,i}$ for each $i\in \{1,\dots, \delta_0\}$. If $r=1$, then $h_0$ is the required homeomorphism. Let $r>1$. Then the set $C^{\mathrm u}_{f^t}\cup C^{\mathrm s}_{f^t}\setminus \mathfrak{l}_0$ is nonempty, and all spheres in this set lie in $\mathfrak{b}_0$. Since the bi-colour graphs are isomorphic, the set $C^{\mathrm s}_{{f'}^t}\cup C^{\mathrm s}_{{f'}^t}\setminus \mathfrak{l}'_0$ is nonempty and all spheres in this set lie in $\mathfrak{b}'_0$. Let $\delta_{i}$ be the degree of the vertex $v_{0,i}$. Since $r>1$, we have $\delta_{i}>1$ for any $i\in \{1,\dots, \delta_0\}$. Let $v_{i,j}$ be a vertex of $\Gamma_{f^t}$ adjacent to $v_{0,i}$ and distinct from $v_0$, $j\in \{1,\dots, \delta_i\}$. Let $l_{i,j}$ be the sphere in $C^{\mathrm u}_{f^t}\cup C^{\mathrm s}_{f^t}\setminus \mathfrak{l}_0$ corresponding to the edge $(v_i, v_{i,j})$, and let $l'_{i,j}$ be the sphere in $C^{\mathrm u}_{{f'}^t}\cup C^{\mathrm s}_{{f'}^t}\setminus \mathfrak{l}'_0$ such that $l'_{i,j}=\xi_*(l_{i,j})$. We set $\mathfrak{l}_{1,i}=\bigcup_{j=1}^{\delta_i}l_{i,j}$ and $\mathfrak{l}'_{1,i}=\bigcup_{j=1}^{\delta_i}l'_{i,j}$, $i\in \{1,\dots, \delta\}$. By the definition of the graph $\Gamma_{f^t}$ the spheres in $\mathfrak{l}_{1,i}$ and $l_{0,i}$ form the boundary of some domain $d_i\subset \mathcal{D}_{f^t}\cap \Sigma_{f^t}$ corresponding to the vertex $v_i$ and contained in the ball $b_{0,i}\subset \mathfrak{b}_0$ bounded by the sphere $l_{0,i}$ (see Figure 11). We denote by $\mathfrak{b}_{1,i}$ the set of balls lying in $\mathfrak{b}_{0,i}$ and bounded by spheres in $\mathfrak{l}_{1,i}$. We have $h_0(\mathfrak{b}_{1,i})\subset {b}'_{0,i}$. By Proposition 3, for each $i\in \{1,\dots,\delta_0\}$ there exists a homeomorphism $h_i\colon \Sigma_{{f'}^t}\to \Sigma_{{f'}^t}$ such that $h_i|_{\Sigma_{{f'}^t}\setminus \operatorname{int}{b}'_{0,i}}=\mathrm{id}$ and $h_i(h_0(l_{1,i}))=l'_{1,i}$ for any sphere $l_{1,i}\in \mathfrak{l}_{1,i}$. If $r=2$, then the composition $h_\delta h_{\delta-1}\dotsb h_1h_0$ is the required homeomorphism $h\colon \Sigma_{{f}^t}\to \Sigma_{{f'}^t}$. Otherwise, we continue the process and, after a finite number of steps, arrive at the required homeomorphism. Case 1, b). The tree $\Gamma_{f^t}$ is bicentral. We denote the sphere corresponding to the central edge of the graph $\Gamma_{f^t}$ by $l_0$. Let $b_0$ be an arbitrary connected component of $\Sigma_{{{f}^t}}\setminus l_0$, and let $v$ be the central vertex corresponding to the domain $d\subset\mathcal{D}_{f^t}\cap \Sigma_{{f}^t}$ lying in $b_0$ (then the boundary of $d$ contains the sphere $l_0$). We denote the sphere corresponding to the central edge of the graph $\Gamma_{{f'}^t}$ by $l'_0$. Let $b'_0$ be the connected component of the set $\Sigma_{{{f'}^t}}\setminus l'_0$ that contains the domain $d'$. By Assertion 6 there exists a homeomorphism $h_0\colon \Sigma_{f^t}\to \Sigma_{{f'}^t}$ such that $h_0(l_0)=l'_0$ and $h_0(b_0)=b'_0$. If ${r=0}$, then $h_0$ 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 $l^{\mathrm s}_{\sigma_1^1},\dots,l^{\mathrm s}_{\sigma_1^g}\in C^{\mathrm s}_{f^t}$ such that the set $\Sigma_{f^t}\setminus \bigcup _{i=1}^g l^{\mathrm s}_{\sigma_1^i}$ is connected. By Lemma 3 the edges $e^{\mathrm s}_1, \dots, e^{\mathrm s}_g$ corresponding to $l^{\mathrm s}_{\sigma_1^1},\dots,l^{\mathrm s}_{\sigma_1^g}$ lie in pairwise different cycles of the graph $\Gamma_{f^t}$. We denote the vertices of $\Gamma_{f^t}$ incident to the edge $e^{\mathrm s}_i$, $i\in \{1,\dots,g\}$, by $v_{i}$ and $ w_i$. Since the graphs $\Gamma_{f^t}$ and $\Gamma_{{f'}^t}$ are isomorphic, the edges ${e'}^{\mathrm s}_1=\xi(e^{\mathrm s}_1)$, $\dots$, ${{e'}^{\mathrm s}_g=\xi(e^{\mathrm s}_g)}$ lie in pairwise different cycles of $\Gamma_{{f'}^t}$. By Proposition 5 the set of spheres ${l'}^{\mathrm s}_{{\sigma'}_{1}^1}=\xi_*(l^{\mathrm s}_{\sigma_1^1})$, $\dots$, ${l'}^{\mathrm s}_{{\sigma'}_{1}^g}=\xi_*(l^{\mathrm s}_{\sigma_1^g})$ corresponding to these edges does not separate the section $\Sigma_{{f'}^t}$. We denote the sets of pairwise disjoint tubular neighbourhoods of spheres in the sets $C^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t}$ and $C^{\mathrm s}_{{f'}^t}\cup C^{\mathrm u}_{{f'}^t}$ by $\{T_l,\, l\in C^{\mathrm s}_{f^t}\cup C^{\mathrm u}_{f^t}\}$ and $\{T_{l'},\, l'\in C^{\mathrm s}_{{f'}^t}\cup C^{\mathrm u}_{{f'}^t}\}$, respectively. By part 3 of Lemma 2, both $\Sigma_{{f}^t}\setminus \bigcup _{i=1}^g \operatorname{int} T_{l^{\mathrm s}_{\sigma_1^i}}$ and $\Sigma_{{f'}^t}\setminus \bigcup _{i=1}^g \operatorname{int} T_{{l'}^{\mathrm s}_{{\sigma'}_1^i}}$ are homeomorphic to a sphere of dimension $n-1$ with $2g$ holes. To the manifold $\Sigma_{{f}^t}\setminus \bigcup _{i=1}^g \operatorname{int} T_{l^{\mathrm s}_{\sigma_1^i}}$ we attach the set $\mathfrak B$ consisting of $2g$ copies of the ball $\mathbb{B}^{n-1}$. Let $\widehat{\Sigma}_{{f}^t}$ be the sphere thus obtained. Let $p_{f^t}$: $\Sigma_{{f}^t}\setminus \bigcup _{i=1}^g \operatorname{int} T_{l^{\mathrm s}_{\sigma_1^i}} \cup \mathfrak B \to \widehat{\Sigma}_{{f}^t}$ be the natural projection. In what follows the sets $p_{f^t}(\Sigma_{{f}^t}\setminus \bigcup _{i=1}^g \operatorname{int} T_{l^{\mathrm s}_{\sigma_1^i}})$, $p_{f^t}(C^{\mathrm s}_{f^t})$, $p_{f^t}(C^{\mathrm u}_{f^t})$, $p_{f^t}(\partial T_{l^{\mathrm s}_{\sigma_1^i}})$ 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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