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This article is cited in 2 scientific papers (total in 2 papers) On classification of Morse–Smale flows on projective-like manifolds V. Z. Grines, E. Ya. Gurevich National Research University "Higher School of Economics"
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2022, Volume 86, Issue 5, DOI: https://doi.org/10.4213/im9197 
Bibliographic databases:
    § 1. IntroductionA flow $f^t$ on a closed smooth manifold $M^n$ of dimension $n\geqslant 1$ is called a Morse–Smale flow if its non-wandering set $\Omega_{f^t}$ consists of finite number of hyperbolic equilibrium states and closed trajectories, and stable and unstable invariant manifolds of the equilibria and closed trajectories either do not intersect each other or intersect transversely. A Morse–Smale flow without closed trajectories is called a gradient-like flow. According to [1], [2], for any gradient-like flow there exists an energy function, that is a Morse function, which strictly decreases along non-singular trajectories of the flow. The set of critical points of the energy function coincides with the set of equilibrium states of the flow, while local minina (local maxima) correspond to sinks (sources), and saddle critical points correspond to the saddle equilibrium states. Moreover, the index of the critical point coincides with the dimension of the unstable invariant manifold of the corresponding equilibrium state.1 Recall that the number $\operatorname{ind}(p)$, equal to the dimension of the unstable manifold of the hyperbolic equilibrium state $p$, is called its Morse index. Since Morse function on any compact manifold has at least one local maximum and one local minimum, then for any gradient-like flow $f^t$ the set $\Omega_{f^t}$ contains at least two equilibria: a source and a sink. If the set $\Omega_{f^t}$ is exhausted by these two points, then it is not difficult to prove that the ambient manifold $M^n$ is homeomorphic to the sphere, and all such flows are topologically equivalent (see, for example, [3], Theorem 2.2.1, where a similar fact for gradient-like cascades was proved, which can be easily adapted for flows). Examples of Morse functions with exactly three critical points (a minimum, a maximum, and a saddle) are constructed in [4], § 19.3, and in [5], § 2. It follows from the results below, that the real and complex projective planes are the only manifolds (up to homeomorphism) of dimension $2$ and $4$, correspondingly, admitting the Morse function with exactly three critical points. Consequently, these manifolds are the only manifolds of dimension $2$ and $4$, admitting gradient-like flows with exactly three equilibria. All such flows (on the same manifolds) are turned to be topologically equivalent, that is not true for similar flows on the manifolds of greater dimension. These results give motivation for obtaining the topological classification of gradient-like flows with an arbitrary non-wandering set on the complex projective plane. The general approach to topological classification of gradient-like flows is based on Poincare–Bendixson and Smale Theorems, that establish the possibility of choosing a finite number of invariant manifolds that cut an ambient manifold into domains with the same asymptotic behavior of trajectories. It seems natural that the mutual arrangement of such invariant manifolds can be described in combinatorial terms, and the isomorphism of the combinatorial invariants is a necessary condition for two flows to be equivalent. As papers of Leontovich, Mayer, Peixoto, Oshemkov, Sharko, Fleitas, Pilyugin, Umanskii show, the isomorphism of combinatorial invariants is not only a necessary but also a sufficient condition of topological equivalence in meaningful classes of gradient-like flows on manifolds. It should be emphasized that all known classification results for flows are obtained under the assumption that any saddle equilibrium state has either stable or unstable manifold of dimension $1$. In this case, the existence of the energy function guarantees that the closures of invariant manifolds which do not participate in heteroclinic intersections, are locally flat (see Proposition 4). However, if the carrying manifold has dimension four and greater, and the set of saddle equilibria contains points with invariant manifolds of codimension two, the closures of these invariant manifolds may be wildly embedded, which makes impossible to obtain complete classification in combinatorial terms. Similar effect was discovered earlier for Morse–Smale cascades on three-dimensional manifolds in papers [6], [7]. In [7], [8] new topological invariants of such cascades was provided and complete topological classification was obtained. We show in Lemma 1 that the non-wandering set of any gradient-like flow without heteroclinic intersections on the complex projective plane contains exactly one saddle with two-dimensional invariant manifolds, moreover, the closures of these manifolds are locally flat spheres. Due to this fact, in the present work it is possible to obtain the complete topological classification of such flows in combinatorial terms. Eels and Kuiper [5] studied manifolds, admitting a Morse function with exactly three critical point. In particular, they obtain the following properties of such manifolds and Morse functions. Statement 1. Let $M^n$ be a connected closed manifold of dimension $n$ and $\varphi\colon M^n\to \mathbb{R}$ be a Morse function, the set of critical points of which consists exactly of three points. Then 1) $n\subset \{2,4, 8, 16\}$; 2) critical points of $\varphi$ have indices $0$, $n/2$, $n$; 3) $M^n$ is the disjoint union of the open ball of dimension $n$ and the sphere of dimension $n/2$; 4) $M^2$ is diffeomorphic to projective plane; 5) for $n\geqslant 4$, $M^n$ is simply connected and orientable; $M^4$ is homotopy equivalent to the complex projective plane. For $n=8$ $(16)$ there exist six (sixty) of homotopy types of such manifolds. Manifolds, admitting a Morse function with exactly three critical points, are called Eels–Kuiper manifolds. Zhuzhoma and Medvedev in [9], [10] showed that for $n=4$ all gradient-like flows whose non-wandering set consists of exactly three points, are topologically equivalent. This implies that for $n=4$ all Eels–Kuiper manifolds are homeomorphic to the complex projective plane. Moreover, due to [9], [10], the topology of Eels–Kuiper manifolds may be refined as follows. Definition 1. A manifold $M^n$ is said to be projective-like if: 1) $n\subset \{2,4, 8, 16\}$; 2) $M^n$ is the disjoint union of the open ball of dimension $n$ and a locally flat sphere2 of dimension $n/2$. According to [10], Theorem 1, a manifold $M^n$ admits a gradient-like flow $f^t$, non-wandering set of which consists exactly of three points if and only if $M^n$ is projective-like. Hence the following statement holds. Indeed, if $M^n$ is projective-like, then it admits a gradient-like flow $f^t$, non-wandering set of which consists of exactly three points. It follows from [1], [2] that the flow $f^t$ has an energy function – a Morse function, the set of critical points of which coincides with the set of equilibria of the flow $f^t$. Then $M^n$ is Eels–Kuiper manifold. On the other hand, if $M^n$ is Eels–Kuiper manifold, then it admits a Morse function with exactly three critical points. A gradient flow of this function is gradient-like, hence, due to [10], Theorem 1, the manifold $M^n$ is projective-like. We denote by $G(M^4)$ a class of gradient-like flows on the complex projective plane $M^4$ such that for any $f^t\in G(M^4)$ invariant manifolds of different saddle equilibria does not intersect each other. We provide necessary and sufficient conditions for flows in $G(M^4)$ to be topological equivalent, and describe an algorithm of realization of all classes of topological equivalence. We now denote by $\Omega^i_{f^t}$ the set of all equilibria of the flow $f^t\in G(M^4)$ with a Morse index $i\in \{0,1,2,3,4\}$. The following statement proved below in § 4 allows to reduce a problem of topological classification of the flows under consideration to a combinatorial problem. Lemma 1. For any $f^t\in G(M^4)$ the following statements hold. 1) If $p \in \Omega^1_{f^t}$ $(p\in \Omega^{3}_{f^t})$, then the closure $\operatorname{cl}W^{\mathrm{s}}_p$ $(\operatorname{cl}W^{\mathrm{u}}_p)$ of the stable (unstable) manifold $W^{\mathrm{s}}_p$ $(W^{\mathrm{u}}_p)$ of $p$ is a locally flat sphere of dimension $3$ that divide $M^4$ into two connected components. 2) The set $\Omega^2_{f^t}$ consists exactly of one equilibrium state, and the closures of its invariant manifolds are locally flat two-dimensional spheres. We are going to show that the classes of topological equivalence of flows in $G(M^4)$ are distinguished by means of bi-color graph, that is defined below similarly to [11]. Denote by $\mathcal{L}_{f^t}$ the set of all spheres $\{\operatorname{cl}W^{\mathrm{s}}_p,\, p\in \Omega_{f^t}^1\}$ and $\{\operatorname{cl}W^{\mathrm{u}}_q,\, q\in \Omega_{f^t}^{3}\}$, and by $k_{f^t}$ the number of these spheres. Since, due to Lemma 1, each sphere from the set $\mathcal{L}_{f^t}$ divides $M^4$ into two connected components, the set $M^4\setminus \bigl(\bigcup_{p\in \Omega_{f^t}^1}\operatorname{cl}W^{\mathrm{s}}_p \cup \bigcup_{q\in \Omega_{f^t}^{n-1}}\operatorname{cl}W^{\mathrm{u}}_q\bigr)$ consists of $m_{f^t}=k_{f^t}+1$ connected components $D_1, \dots, D_{m_{f^t}}$. Denote by $\mathcal{D}_{f^t}$ the set of all these components. Definition 2. A bi-color graph of a flow $f^t\,{\in}\, G(M^4)$ is a graph $\Gamma_{f^t}$, defined as follows (see fig. 1): 1) the set $V(\Gamma_{f^t})$ of the vertices of the graph $\Gamma_{f^t}$ is isomorphic to the set $\mathcal{D}_{f^t}$, the set $E(\Gamma_{f^t})$ of the edges of the graph $\Gamma_{f^t}$ is isomorphic to the set $\mathcal{L}_{f^t}$; 2) the vertices $v_i$, $v_j$ are incident to an edge $e_{i, j}$ if and only if the corresponding domains $D_i, D_j$ have a common boundary component; 3) the edge $e_{i, j}$ has the color $\mathrm s$ $(\mathrm u)$ if it corresponds to a manifold $\operatorname{cl}W^{\mathrm{s}}_p\in \mathcal{L}_{f^t}$ ($\operatorname{cl}W^{\mathrm{u}}_q \in \mathcal{L}_{f^t}$); 4) the graph $\Gamma_{f^t}$ has a unique marked vertex $v_*$ corresponding to the domain $D_*\subset\mathcal{D}_{f^t}$ which contains the saddle equilibrium state whose Morse index equals $2$. Definition 3. Graphs $\Gamma_{f^t}, \Gamma_{g^t}$ of flows $f^t, g^t\in G(M^4)$ are isomorphic if there exists an isomorphism $\xi\colon \Gamma_{f^t}\to \Gamma_{g^t}$ preserving the colors of the edges and the marked vertex. Recall that a tree is a connected graph such that for any pair of its vertices there exists a single path connecting these vertices. We prove in § 6 that a bi-color graph of any flow $f^t\in G(M^4)$ is the tree. The main results of the paper are the following. Theorem 1. Flows $f^t, g^t\in G(M^4)$ are topologically equivalent if and only if their bi-color graphs $\Gamma_{f^t}$, $\Gamma_{g^t}$ are isomorphic. Theorem 2. For any tree $\Gamma$ with a marked vertex $v_*\in V(\Gamma)$ whose edges are arbitrarily colored in two colors, there exists a flow $f^t\in G(M^4)$ whose bi-color graph $\Gamma_{f^t}$ is isomorphic to $\Gamma$. § 2. Auxiliary topological factsThis section contains auxiliary topological facts used in the paper. 2.1. Homotopy groups of the ambient manifoldThe following statement clarifies properties of the complex projective plane. Proposition 1. Let $M^4$ be the complex projective plane. Then homotopy groups $\pi_1(M^4)$, $\pi_3(M^4)$, $\pi_4(M^4)$ are trivial and groups $\pi_2(M^4)$, $\pi_5(M^4)$ are isomorphic to $\mathbb{Z}$. We preliminarily give all the definitions and facts necessary for the proof. Let $X$, $Y$ be topological spaces. Continuous maps $h,g\colon X\to Y$ are said to be homotopic if there exists a continuous map (a homotopy) $H\colon X\times [0,1]\to Y$ such that $H|_{X\times \{0\}}=h, H|_{X\times \{1\}}=g$. A homotopy $H\colon X\times [0,1]\to Y$ is often presented as a family $h_t\colon X\to Y$ of maps such that $H(x,t)=h_t(x)$, $x\in X$, $t\in [0,1]$. Let $I^i\subset \mathbb{R}^i$, $i\geqslant 1$, be a unit cube, $\partial I^i$ be its boundary, and let $f\colon I^i\to X$ be a continuous map such that $f(\partial I^i)=x_0$, where $x_0\in X$ is a point in $X$. Denote by $[f]$ a class of maps homotopic to $f$ such that every map and the homotopy maps $\partial I^i$ to $x_0$. A set of equivalent classes $[f]$, $[g]$ with the operation $*$ given by the rule
$$
\begin{equation*}
[f]*[g]=[f*g]=\begin{cases} f(2t_1,t_2,\dots,t_i), &0\leqslant t_1\leqslant \dfrac12, \\ g(2t_1-1,t_2,\dots,t_i), &\dfrac12\leqslant t_1\leqslant 1, \end{cases}
\end{equation*}
\notag
$$
is called an $i$-dimensional homotopy group and is denoted by $\pi_i(X, x_0)$. A group $\pi_1(X, x_0)$ is called a fundamental group. A bundle is a triple $(E,p,B)$, where $E$ is a topological space, called the total space, $B$ is another topological space, called the base space of the bundle, and $p\colon E\to B$ is a surjective map called the projection. The map $p$ itself is often called a bundle. For each element $b\in B$, the fibre over this element is defined as a subset $F_{b}\subset E$ of all pre-images of the element $b$. A bundle $(E,p,B)$ is called a fiber bundle if for any point $b\in B$ there is a neighborhood $U_b\subset B$ and a homeomorphism $\varphi\colon p^{-1}(U_b)\to U_b\times F_b$ such that $p|_{p^{-1}(U_b)}= \operatorname{proj}_1 \varphi|_{p^{-1}(U_b)}$, where $\operatorname{proj}_1\colon U_b\times F_b\to U_b$ is a projection defined by $\operatorname{proj}_1(x,y)=x$, $x\in U_b$, $y\in F_b$. Let $(E,p, B)$ be a fiber bundle with a fiber $F$ and a connected base space. Fix a point $b_0\in B$. The map $p\colon X\to B$ induces a homomorphism $p_* \colon \pi_i(E, e_0)\to \pi_i(B, b_0)$, where $e_0\in F_{b_0}$. The inclusion $i\colon F_b\to E$ induces a homomorphism $i_*\colon \pi_i(F_b, e_0)\to \pi_i(E, e_0)$. For $i\geqslant 1$, we define a map $\partial_*\colon \pi_{i}(B, b_0)\to \pi_{i-1}(F_b, e_0)$, which is a homomorphism for $i\geqslant 2$, in the following way (see. [4], Ch. 4, § 14.3). The map $f\colon (I^i, \partial I^i)\to (B, b_0)$ is a homotopy, connecting constant maps $\varphi_0,\varphi_1\colon \partial I^i\to b_0$. According to Covering Homotopy Theorem, there exists a homotopy $\widetilde{\varphi}_t\colon \partial I^i\to E$ such that $\widetilde{\varphi}_0(\partial I^i)=e_0$ and $p \widetilde{\varphi}_t=\varphi_t$. Since $\varphi_1(\partial I^{i})=b_0$, $\widehat{\varphi}_1(\partial I^i)\subset F_{b_0}$. We note that the set $\partial I^{i}$ is homeomorphic to the cube $I^{i-1}$ with the boundary $\partial I^{i-1}$ contracted to a point. Then define the image $\partial_*f$ of the map $f$ as the homotopy class of the map $\widehat{\varphi}_1\colon (I^{i-1}, \partial I^{i-1})\to (F, e_0)$. A sequence
$$
\begin{equation*}
\cdots G_{i}\xrightarrow{\varphi_{i}}G_{i-1}\xrightarrow{\varphi_{i-1}} G_{i-2}\cdots
\end{equation*}
\notag
$$
of the groups and homomorphisms is called an exact sequence if for any $i$ the image $\operatorname{Im}_{\varphi_{i}}=\{\varphi_{i}(g),\, g\in G_{i}\}$ of the homomorphism $\varphi_{i}$ coincides with the kernel $\operatorname{Ker}_{\varphi_{i-1}}=\{g\in G_{i-1}\colon \varphi_{i-1}(g)=0\}$ of the homomorphism $\varphi_{i-1}$. The proof of the following statement can be found, for example, in [4], Theorem 14.3. Statement 3. Let $(E,p, B)$ be a fiber bundle with a fiber $F$. Then the following sequence is exact: Recall that a sphere $S^k$ of dimension $k\geqslant 0$ is the manifold homeomorphic to the unit sphere
$$
\begin{equation*}
\mathbb{S}^k=\{(x_1,\dots,x_{k+1})\subset \mathbb{R}^{k+1}\mid x_1^2+\dots+x_{k+1}^2=1\}.
\end{equation*}
\notag
$$
A ball (an open ball) $B^n$ of dimension $n\geqslant 1$ is the manifold homeomorphic to the unit ball (the interior of the unit ball)
$$
\begin{equation*}
\mathbb{B}^n=\{(x_1,\dots,x_n)\subset \mathbb{R}^n\mid x_1^2+\dots+x_n^2\leqslant 1\}.
\end{equation*}
\notag
$$
A canonical complex plane is the set $\mathbb{C}P^2$ of all equivalence classes $[a,b,c]$ of triples $(a,b,c) \in \mathbb{C}^3\setminus (0,0,0)$ with respect to equivalence relation $(a,b,c)\sim (a',b',c')$ if $(a,b,c)=(\lambda a',\lambda b', \lambda c')$, where $\lambda$ is a non-zero complex number. Consider the unit five-dimensional sphere as the subset $S^5\,{=}\,\{(a,b,c)\in \mathbb{C}^3\mid |a|^2+|b|^2+|c|^2=1\}$ of the space $\mathbb C^3$ and identify points of the sphere in the equivalence relation $(a,b,c)\sim (\lambda a',\lambda b', \lambda c')$, $|\lambda|=1$. A quotient space $S^5/_{\sim}$ under this equivalence relation if homeomorphic to $\mathbb C P^2$. According to [4], Ch. 4, § 14.3, the following statement holds. Statement 4. A canonical projection $p\colon S^5\to \mathbb C P^2$ is a fiber bundle with a fiber homeomorphic to the circle. We use the exact sequence of the fiber bundle $p\colon S^5\to \mathbb C P^2$ to prove Proposition 1. Proof of Proposition 1. Consider the following subsequence of the exact sequence of the fiber bundle $p\colon S^5\to \mathbb C P^2$: The group $\pi_0(S^1)$ coincides with the number of connected components of $S^1$ and, consequently, is trivial, hence $\operatorname{Ker}_{\partial_*}=\pi_1(\mathbb{C}P^2)$. Since the sequence is exact, $\operatorname{Ker}_{\partial_*}=\operatorname{Im}_{p_*}$, therefore $\pi_1(\mathbb{C}P^2)=\operatorname{Im}_{p_*}$ and $p_*$ is an isomorphism. Since $\pi_1(S^5)$ is trivial, $\pi_1(\mathbb{C}P^2)$ is also trivial.
Let us prove that $\pi_3(\mathbb C P^2), \pi_4(\mathbb C P^2)$ are trivial. Consider exact subsequences
$$
\begin{equation*}
\begin{gathered} \, \pi_{3}(S^5)\xrightarrow{p_*} \pi_{3}(\mathbb{C}P^2)\xrightarrow{\partial_*} \pi_{2}(S^1), \\ \pi_{4}(S^5)\xrightarrow{p_*} \pi_{4}(\mathbb{C}P^2)\xrightarrow{\partial_*} \pi_{3}(S^1), \end{gathered}
\end{equation*}
\notag
$$
and apply the similar arguments taking into account that the groups $\pi_i(S^1)$ for $i\geqslant 2$ and $\pi_j(S^5)$ for $0\leqslant j<5$ are trivial.
Consider an exact subsequence
$$
\begin{equation*}
\pi_{2}(\mathbb{C}P^2)\xrightarrow{\partial_*} \pi_1(S^1)\xrightarrow{p_*} \pi_1(\mathbb{C}P^2).
\end{equation*}
\notag
$$
Since $\pi_1(\mathbb{C}P^2)$ is trivial, the map $\partial_{*}\colon\pi_{2}(\mathbb{C}P^2)\to\pi_1(S^1)$ is an isomorphism. Since $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$, $\pi_2(\mathbb{C}P^2)$ is also isomorphic to $\mathbb{Z}$.
Similarly, due to exactness of the sequence and the fact that $\pi_{5}(S^5)$ is isomorphic to $\mathbb{Z}$, one gets that $\pi_{5}(\mathbb{C}P^2)$ is isomorphic to $\mathbb{Z}$. The Proposition is proved.2.2. Dehn SurgeryA solid torus is the manifold $\Pi$ homeomorphic to the direct product $\mathbb{S}^1\times\mathbb{B}^2$ of the circle and the two-dimensional disk. Let $x\in \mathbb{S}^1$, $y\in \partial\mathbb B^2$ be arbitrary points and $\varphi\colon \mathbb{S}^1\times \mathbb{B}^2\to \Pi$ be a homeomorphism. Curves
$$
\begin{equation*}
m=\varphi(\{x\}\times \partial \mathbb B^2),\qquad l=\varphi(\mathbb{S}^1\times \{y\})
\end{equation*}
\notag
$$
are called the canonical meridian and longitude, respectively, of the solid torus $\Pi$. We say that curves $\mu, \lambda$ lying in the boundary torus are meridian and longitude if they are homotopic to $m$, $l$, correspondingly. Since the fundamental group $\pi_1(T)$ of the torus is isomorphic to the group $\mathbb{Z}\times \mathbb{Z}$, the homotopy class $[\gamma]$ of any oriented closed curve $\gamma\in T$ is defined by a pair of co-prime integers $(p,q)$. We assume that $[l]=(1,0)$, $[m]=(0,1)$.According to [13], Ch. 2, C, Theorem 4, the following statement holds. Statement 5. Let $\psi\colon \partial \Pi\to \partial \Pi$ be a homeomorphism. A homeomorphism $\Psi\colon \Pi\to \Pi$ such that $\Psi|_{\partial \Pi}=\psi|_{\partial \Pi}$ exists if and only if $\psi$ preserves the homotopy class of the meridian. Due to [13], Ch. 2, B, C, a homeomorphism $\psi\colon T\to T$ induces a homomorphism $\psi_*\colon\pi_1(T)\to \pi_1(T)$ which is uniquely determined by an unimodular integer matrix $A_{\psi}$ such that $[\psi(\gamma)]=(p,q)A_{\psi}$ for $[\gamma]=(p,q)$. According to [13], Ch. 2, C, Theorem 4, the following statement holds. Statement 6. Homeomorphisms $\psi, \psi'\colon T\to T$ are isotopic if and only if $A_{\psi}\,{=}\, A_{\psi'}$. Corollary 1. A homeomorphism $\psi\colon T\to T$ is isotopic to the identity map $\operatorname{id}\colon T\to T$ if and only if $\psi$ preserves the homotopy classes of the meridian and longitude. Proof. Set
Then $[\psi(1,0)]=(1,0)A_\psi=(a, b)$, $[\psi(0,1)]=(0,1)A_\psi=(c,d)$. Hence the homeomorphism $\psi$ preserves the homotopy classes of the meridian and longitude if and only if the matrix $A_\psi$ is unite. Hence, due to Statement 6, homeomorphism $\psi$ is isotopic to identity. The Corollary is proved. Recall that a simple closed curve $\mathcal{C}\subset {S}^3$ is called a knot. A knot $\mathcal{C}\subset S^3$ is said to be trivial if there exists a homeomorphism $\varphi\colon S^3\to\mathbb{S}^3$ such that
$$
\begin{equation*}
\varphi(\mathcal{C})=\{(x_1,x_2,x_3, x_4)\in \mathbb{R}^4\colon x_1^2+x_2^2=1,\, x_3=x_4=0\}.
\end{equation*}
\notag
$$
Let $\mathcal C\subset S^3$ be a knot and $\Pi_{\mathcal C}\subset S^3$ its tubular neighborhood. Dehn Surgery along $\mathcal C\subset S^3$ is a way of obtaining a new closed manifold by gluing two manifolds with boundary $S^3\setminus \operatorname{int}\Pi_{\mathcal C}$ and $\mathbb{S}^1\times\mathbb{B}^2$ by means of a homeomorphism $\varphi\colon \partial (\mathbb{S}^1\times \mathbb{B}^2) \to \partial \Pi_{\mathcal C}$. Let $m\subset \partial (\mathbb{S}^1\times \mathbb{B}^2)$ be a canonical meridian and $[\phi(m)]=(p,q)$. The ratio $q/p$ is called the surgery coefficient. A surgery with zero coefficient is said to be trivial. It follows from the Statement 5 that trivial surgery does not change the topology of the manifold. By [14], Theorem 2, the following statement is true. 2.3. Local homeomorphisms extendingDue to [15], Ch. 8, Theorem 3.1, 3.2, the following statement holds. Statement 8. Let $e_i, e'_i\colon \mathbb{B}^n\to \operatorname{int}\mathbb{B}^n$, $i\in \{1,\dots,k\}$, be smooth embeddings such that $e_i(\mathbb{B}^n)\cap e_j(\mathbb{B}^n)=\varnothing$, $e'_i(\mathbb{B}^n)\cap e'_j(\mathbb{B}^n)=\varnothing$ for any $i,j\in \{1,\dots,k\}$, $i\neq j$. Then there exists a homeomorphism $h\colon \mathbb{B}^n\to \mathbb{B}^n$ such that 1) $h|_{\partial \mathbb{B}^n}=\operatorname{id}$; 2) $he_i=e_i'$, $i\in \{1,\dots,k\}$. Proposition 2. Let $C \subset \mathbb{S}^3$ be a trivial knot, $\Pi_0\subset \mathbb{S}^3$ be its tubular neighborhood, and let $e, e'\colon \mathbb{S}^1\times \mathbb{B}^2\to \operatorname{int} \Pi_0$ be topological embeddings such that $e (\mathbb{S}^1\times \{x\})= e'(\mathbb{S}^1\times \{x\})$, $x\in \partial \mathbb{B}^2$. Then there exists a homeomorphism $\theta\colon \mathbb{S}^3\to \mathbb{S}^3$ such that 1) $\theta|_{\mathbb{S}^3\setminus \operatorname{int} \Pi_0}=\operatorname{id}$; 2) $\theta e=e'$. Proof. Since $e$, $e'$ are the embeddings of the solid tori, due to Statement 5 the homeomorphism ${e'}^{-1}e\colon \mathbb{S}^1\times \mathbb{B}^2\to \mathbb{S}^1\times \mathbb{B}^2$ maps the meridian of solid torus $\mathbb{S}^1\times \mathbb{B}^2$ into a curve homotopic to it. It follows from the condition of the Proposition, that the homeomorphism ${e'^{-1}}e$ maps the homotopy class of longitude of $\mathbb{S}^1\times \mathbb{B}^2$ into itself. Then, according to Corollary 1, there exists an isotopy such that
Set
$$
\begin{equation*}
\mathbf{\Pi}=(\mathbb{S}^1\times \mathbb{B}^2)\bigcup_{\varphi} (\mathbb{S}^1\times \mathbb{S}^1\times [0,1]),
\end{equation*}
\notag
$$
where $\varphi\colon \partial \mathbb{S}^1\times \mathbb{B}^2 \to \mathbb{S}^1\times \mathbb{S}^1\times \{0\}$ is the identity homeomorphism.
Define a homeomorphism $H\colon \mathbf{\Pi}\to \mathbf{\Pi}$ acting identically on the set $\mathbb{S}^1\times \mathbb{B}^2$ and equal to
$$
\begin{equation*}
H(x,t)=(h_t(x),t),\qquad x\in \mathbb{S}^1\times \mathbb{S}^1,\quad t\in [0,1],
\end{equation*}
\notag
$$
for points $(x,t)$ from the set $\mathbb{S}^1\times \mathbb{S}^1\times [0,1]$.
Set $\Pi=e(\mathbb{B}^2\times \mathbb{S}^1)$, $\Pi'=e'(\mathbb{B}^2\times \mathbb{S}^1)$. It follows from [16], Theorem 3.3, that manifolds $\Pi_0\setminus \operatorname{int} \Pi$, $\Pi_0\setminus \operatorname{int} \Pi'$ are homeomorphic to the direct product of the torus and the interval. Then there are two foliations $\{T_q\}$, $\{I_p\}$ on $\Pi_0\setminus \operatorname{int} \Pi$ with leaves homeomorphic to the torus $\mathbb{S}^1\times \mathbb{S}^1$ and the segment $[0,1]$, respectively, such that connected components of the boundary of the manifold $\Pi_0\setminus \operatorname{int} \Pi$ belongs to the foliation $\{T_q\}$ and for every pair of leaves $T_q$, $I_p$ the intersection $T_q\cap I_p$ is non-empty and consists of a single point. Assume that the leaves of the foliation $\{T_q\}$ are parametrized by points $q\in [0,1]$, and the leaves of the foliation $\{I_p\}$ are parametrized by points $p\in\partial \Pi=T_0$. Hence, any point from the set $\Pi_0\setminus \operatorname{int} \Pi$ is determined by two coordinates $(p,q)$, $p\in \partial \Pi$, $q\in [0,1]$. Define an embedding $\widetilde{e}\colon \mathbb{S}^1\times \mathbb{S}^1\times [0,1]\to \Pi_0\setminus \operatorname{int} \Pi$ by
$$
\begin{equation*}
\widetilde{e}(x,t)=(e(x)|_{\partial (\mathbb{S}^1\times \mathbb{B}^2)}, t),\qquad x\in \mathbb{S}^1\times \mathbb{S}^1.
\end{equation*}
\notag
$$
The embedding $e\colon \mathbb{B}^2\times \mathbb{S}^1\to \Pi$ may be extended up to the embedding $E\colon \mathbf{\Pi} \to \Pi_0$ by
$$
\begin{equation*}
E(z)=\begin{cases} e(z), &z\in \mathbb{B}^2\times \mathbb{S}^1, \\ \widetilde{e}(z), &z\in \mathbb{S}^1\times \mathbb{S}^1\times [0,1]. \end{cases}
\end{equation*}
\notag
$$
Similarly, extend the embedding $e'\colon \mathbb{B}^2\times \mathbb{S}^1\to \Pi'$ up to the embedding $E'\colon \mathbf{\Pi}\to \Pi_0$. Lastly, set The map $\theta\colon \mathbb{S}^3\to \mathbb{S}^3$ is the desired homeomorphism. The proposition is proved.Proposition 3. Let $S^2\subset \mathbb{S}^3$ be a smoothly embedded two-dimensional sphere, $A_0\subset \mathbb{S}^3$ be its tubular neighborhood, and let $e, e'\colon \mathbb{S}^2\times [0,1]\to \operatorname{int} A_0$ be topological embeddings. Then there exists a homeomorphism $\theta\colon \mathbb{S}^3\to \mathbb{S}^3$ such that 1) $\theta|_{\mathbb{S}^3\setminus \operatorname{int} A_0}=id$, 2) $\theta e=e'$. Proof. Set $A= e(\mathbb{S}^2\times [0,1])$, $A'=e'(\mathbb{S}^2\times [0,1])$. A homeomorphism $e'^{-1}e|_{\mathbb{S}^1\times \{0\}}$ is an orientation preserving homeomorphism of the sphere, hence it is isotopic to the identity map. It follows from the Annulus Theorem that each of the set $A_0\setminus \operatorname{int} A$, $A_0\setminus \operatorname{int} A'$ has two connected components homeomorphic to the direct product $\mathbb{S}^2\times [0,1]$. Further constructions is similar to those from the proof of Proposition 2. The proposition is proved.
§ 3. Auxiliary facts characterizing the dynamics of gradient-like flowsIn this section, we present a number of important auxiliary facts that help to describe the dynamics of flows from the class under consideration in a canonical way. The following statement follows from [1], Theorem B, [2], Theorem 1. Statement 9. For any gradient-like flow $f^t$ on a closed manifold $M^n$ there exists a Morse function $\varphi \colon M^n\to [0,n]$ such that 1) the set of the critical points of the function $\varphi$ coincides with the set $\Omega_{f^t}$; 2) $\varphi(f^t(x))<\varphi(x)$ for any point $x\not\in \Omega(f^t)$ and for any $t>0$; moreover, all non-closed trajectories of the flow intersect the level sets of the function $\varphi$ transversally; 3) $\varphi(p)=\operatorname{ind}(p)$ for any $p\in \Omega_{f^t}$; 4) in a neighborhood of any point $p\in \Omega^i_{f^t}$ there exist local coordinates3 $y_1,\dots,y_n$ such that the function $\varphi$ can be presented as $\varphi=\varphi(p)-y_1-\dots-y^2_i+y^2_{i+1}+\dots+y^2_n$. The statement below follows from [18], Theorem 2.3. Statement 10. Let $f^t$ be a gradient-like flow on a closed manifold $M^n$, $n\geqslant 1$. Then 1) $M^n=\bigcup_{p\in \Omega_{f^t}}W^{\mathrm{s}}_p= \bigcup_{p\in \Omega_{f^t}}W^{\mathrm{u}}_p$; 2) for any point $p\in \Omega_{f^t}$ the manifold $W^{\mathrm{u}}_p$ is a smooth submanifold of the manifold $M^n$; 3) for any point $p\in \Omega_{f^t}$ and any connected component $l^{\mathrm{u}}_p$ of the set $W^{\mathrm{u}}_p\setminus p$ the equality $ \operatorname{cl}l^{\mathrm{u}}_p\setminus (l^{\mathrm{u}}_p \cup p)= \bigcup_{q \in \Omega_{f}\colon W^{\mathrm{s}}_q \cap l^{\mathrm{u}}_q \neq \varnothing}W^{\mathrm{u}}_q$ holds. From item 1) of the Statement 10 one can easily obtain the following fact. Corollary 2. The non-wandering set $\Omega_{f^t}$ of any gradient-like flow $f^t$ contains at least one sink and one source. Proof. Assume the contrary. For definiteness, we assume that the set of source equilibrium states is empty. By Part 1) of Statement 10, the manifold $M^n$ is represented as a union of unstable manifolds of equilibrium states. Since the equilibrium states of the flow $f^t$ are hyperbolic, the unstable manifold of the equilibrium state $p$ is a ball of dimension $k=\{0,\dots,n\}$. Thus, in the absence of sources, a manifold $M^n$ of dimension $n$ can be represented as a finite union of smoothly embedded balls of dimension less than $n$, which is impossible. The corollary is proved. Proposition 4. Let $f^t$ be a gradient-like flow on the closed manifold $M^n$, $n\geqslant 3$, and $\sigma$ be a saddle equilibrium state of the flow $f^t$ such that 1) Morse index of the saddle $\sigma$ equals $(n-1)$; 2) $W^{\mathrm{u}}_{\sigma}\cap W^{\mathrm{s}}_{\sigma'}= \varnothing$ for any saddle $\sigma'\neq \sigma$. Then $\operatorname{cl} W^{\mathrm{u}}_{\sigma}$ is a locally flat sphere of dimension $(n-1)$. Proof. By assumption, $W^{\mathrm{u}}_{\sigma}\cap W^{\mathrm{s}}_{\sigma'}= \varnothing$ for any saddle equilibrium $\sigma'\neq \sigma$, hence, due to Part 3) of Statement 10, there exists a sink equilibrium state $\omega$ such that $\operatorname{cl}W^{\mathrm{u}}_\sigma=W^{\mathrm{u}}_{\sigma}\cup \omega$. Therefore, $\operatorname{cl}W^{\mathrm{u}}_\sigma$ is a sphere of dimension $(n-1)$ topologically embedded in $M^n$. According to Part 2) of Statement 10 the sphere $\operatorname{cl}W^{\mathrm{u}}_\sigma$ is smooth, and, consequently, locally flat at all points of $W^{\mathrm{u}}_\sigma$.
We prove that the sphere $\operatorname{cl}W^{\mathrm{u}}_\sigma$ is locally flat at the point $\omega$. Let $\varphi\colon M^n\to \mathbb{R}$ be the energy function of the flow $f^t$ defined by Statement 9, $\varepsilon\in (0,1)$, and $\Sigma_\omega$ be a connected component of the set $\varphi^{-1}(\varepsilon)$ that belongs to $W^{\mathrm{s}}_{\omega}$. It follows from Part 4) of Statement 9 that $\Sigma_\omega$ is a smooth sphere of dimension $(n-1)$. Since $W^{\mathrm{u}}_\sigma\setminus \sigma\subset W^{\mathrm{s}}_\omega$, there exists a smoothly embedded $(n-1)$-ball $B^{\mathrm{u}}\subset W^{\mathrm{u}}_{\sigma}$ such that $\sigma\in \operatorname{int} B^{\mathrm{u}}$ and $\Sigma_{\omega}\cap (W^{\mathrm{u}}_{\sigma}\setminus B^{\mathrm{u}})=\varnothing$. Then $W^{\mathrm{u}}_\sigma\cap \Sigma_\omega=B^{\mathrm{u}}\cap \Sigma_{\omega}$. Since the function $\varphi$ strictly decreases along non-singular trajectories of the flow $f^t$, the intersection $B^{\mathrm{u}}\cap \Sigma_{\omega}$ is transversal. Hence the intersection
$$
\begin{equation*}
X^{\mathrm{u}}_{\sigma}=W^{\mathrm{u}}_{\sigma}\cap \Sigma_{\omega}
\end{equation*}
\notag
$$
is a smooth closed manifold of dimension $n-1+n-1-n\,{=}\,n-2$. Since $\Sigma_{\omega}$ is the section for trajectories of the flow $f^t$, the submanifold $X^{\mathrm{u}}_{\sigma}$ is the section for all trajectories of the restriction of the flow $f^t$ on $W^{\mathrm{u}}_\sigma\setminus \sigma$. Define a retraction
$$
\begin{equation*}
r\colon W^{\mathrm{u}}_\sigma\setminus \sigma\to X^{\mathrm{u}}_{\sigma},
\end{equation*}
\notag
$$
by putting into the correspondence to a point $x\in W^{\mathrm{u}}_\sigma\setminus \sigma$ the point $y=f^{t_x}(x)\in X^{\mathrm{u}}_{\sigma}$ of the intersection of the trajectory of the flow $f^t$ with the set $X^{\mathrm{u}}_{\sigma}$ passing through the point $x$. The retraction $r$ can be connected with the identity map $\operatorname{id}\colon W^{\mathrm{u}}_\sigma\setminus \sigma\to W^{\mathrm{u}}_\sigma\setminus \sigma$ by a homotopy therefore, $X^{\mathrm{u}}_{\sigma}$ is homotopically equivalent to $W^{\mathrm{u}}_\sigma\setminus \sigma$. Since $W^{\mathrm{u}}_{\sigma}\setminus \sigma$ is homeomorphic to $\mathbb{R}^{n-1}\setminus O$, it is homotopy equivalent to the sphere of dimension $(n-2)$. Then, due to Poincare Conjecture which was proved for all dimensions, the manifold $X^{\mathrm{u}}_{\sigma}$ is homeomorphic to the $(n-2)$-sphere.
It follows from the generalized Schoenflies Theorem (see [19], [20]) that there exists a homeomorphism $\psi_{\mathrm{u}}\colon \Sigma_{\omega}\to \mathbb{S}^{n-1}$ such that $\psi_{\mathrm{u}}(X^{\mathrm{u}}_{\sigma})=\mathbb{S}^{n-2}=\mathbb{S}^{n-1}\cap Ox_1\dots x_{n-1}$. Denote by $V_{\omega}$ an $n$-ball laying in $W^{\mathrm{s}}_\omega$ and bounded by $\Sigma_{\omega}$. We construct a homeomorphism $\Psi_{\mathrm{u}}\colon V_\omega\to \mathbb{B}^n$ conjugated the flow $f^t|_{V_\omega}$ with the linear flow $a^t\colon \mathbb{R}^n\to \mathbb{R}^n$, $a(x_1,\dots, x_n)=((1/2)^t x_1, \dots, (1/2)^t x_n)$. To do this, we assign to each point $x\in V_\omega\setminus \omega$ a time $t_x\leqslant 0$ such that $f^{t_x}(x)\subset \Sigma_\omega$ and set
$$
\begin{equation*}
\Psi_{\mathrm{u}}(x)=\begin{cases} a^{-t_x}(\psi_{\mathrm{u}}(f^{t_x}(x))), &x\in V_{\omega}\setminus \omega, \\ O, &x=\omega. \end{cases}
\end{equation*}
\notag
$$
By construction, $\Psi_{\mathrm{u}}(\operatorname{cl}W^{\mathrm{u}}_\sigma\cap V_\omega)\subset Ox_1\dots x_{n-1}$. Then the sphere $\operatorname{cl}W^{\mathrm{u}}_{\sigma}$ is locally flat at the point $\omega$. The proposition is proved. Replacing the flow $f^t$ by $f^{-t}$ in Proposition 4, one can immediately obtain the following statement. Corollary 3. Let $f^t$ be a gradient-flow on a closed manifold $M^n$, $n\geqslant 3$, and $\sigma$ be a saddle equilibrium state of the flow $f^t$ such that 1) Morse index of $\sigma$ equals $1$; 2) $W^{\mathrm{s}}_{\sigma}\cap W^{\mathrm{u}}_{\sigma'}= \varnothing$ for any saddle $\sigma'\neq \sigma$. Then $\operatorname{cl} W^{\mathrm{s}}_{\sigma}$ is a locally flat sphere of dimension $(n-1)$. The following statement defines a canonical neighborhood $V_p$ of a saddle hyperbolic equilibrium $p$, enriched by a coordinate system, in which the flow $f^t$ can be represent as a product. The proof of this statement can be found in [21], Ch. 2, § 7, Lemmas 7.2, 7.3. Statement 11. Let $p$ be a hyperbolic equilibrium of the flow $f^t\colon M^n\to M^n$ and $i\in \{1,\dots,n-1\}$ be the Morse index of $p$. Then there exists a compact neighborhood $V_p\subset M^n$ of the point $p$ enriched with two continuous maps $\pi_{\mathrm{s}}\colon V_p\to B^{\mathrm{s}}_p, \pi_{\mathrm{u}}\colon V_p\to B^{\mathrm{u}}_p$, where $B^{\mathrm{s}}_p=V_p\cap W^{\mathrm{s}}_p$, $B^{\mathrm{u}}_p=V_p\cap W^{\mathrm{u}}_p$ are balls of dimensions $(n- i)$, $i$, respectively, containing the point $p$, with the following properties (Fig. 2): 1) $\pi_{\mathrm{s}}^{-1}(p)=B^{\mathrm{u}}_p$, $\pi_{\mathrm{u}}^{-1}(p)=B^{\mathrm{s}}_p$; 2) for any point $x\in B^{\mathrm{s}}_p$ ($y\in B^{\mathrm{u}}_p$) its preimage $\pi_{\mathrm{s}}^{-1}(x)$ $(\pi_{\mathrm{u}}^{-1}(y))$ is a smoothly embedded in $M^n$ ball of dimension $i$ $(n-i)$; 3) for any points $x\in B^{\mathrm{s}}_p$, $y\in B^{\mathrm{u}}_p$ balls $\pi_{\mathrm{s}}^{-1}(x)$, $\pi_{\mathrm{u}}^{-1}(u)$ intersect each other at a single point; 4) $f^t(\pi_{\mathrm{s}}^{-1}(x))\supset \pi_{\mathrm{s}}^{-1}(f^t(x))$, $f^{-t}(\pi_{\mathrm{u}}^{-1}(y)\supset \pi_{\mathrm{u}}^{-1}(f^{-t}(y))$ for any $t\geqslant 0$. Remark 1. If $f^t$ is a gradient-like flow then it is possible to choose the neighborhoods $V_p$ of its saddle equilibria in such a way that $V_p\cap V_q=\varnothing$ for any $p\neq q$. We assume below that the canonical neighborhoods have this property. As a corollary of Statement 11 we obtain a classical fact that any flow in a neighborhood of a hyperbolic equilibria is locally topologically conjugated with a linear flow. Note that this statement is a reformulation of the classical Grobman–Hartman theorem (see [22], [23], and also [21], Ch. 2, § 7, Proposition 2.15). We provide a version of its proof, focusing on the technical fact that the conjugating homeomorphism can be defined on the compact canonical neighborhood $V_p$. Let $b^t_i$ be a linear flow on the space $\mathbb{R}^n=\mathbb{R}^{n-i}\times \mathbb{R}^i$ given by
$$
\begin{equation*}
b^t_i(x,y)=(e^{-t}x,e^ty), \qquad x\in \mathbb{R}^{n-i},\quad y\in \mathbb{R}^{i}.
\end{equation*}
\notag
$$
The flow $b^t_i$ has a unique equilibrium state at the origin $O$, this equilibrium state is hyperbolic and its Morse index equals $i$. Set $\mathbb{B}^{n-i}=\{x\in \mathbb{R}^{n-i}\colon |x|\leqslant 1\}$, $\mathbb{B}^i=\{y\in \mathbb{R}^i\colon |y|\leqslant 1\}$. Proposition 5. Let $p$ be an equilibrium of a flow $f^t\colon M^n\to M^n$ and Morse index of $p$ equals $i\in \{1,\dots,n\}$. Then there exists a homeomorphism $h_p\colon V_p\to \mathbb{B}^{n-i}\times \mathbb{B}^i$ such that $h_pf^t=b^t_ih_p$ for any $t\in \mathbb{R}$ for which the right and left sides are defined. Proof. Let $B^{\mathrm{s}}_p\subset W^{\mathrm{s}}_p$, $B^{\mathrm{u}}_p\subset W^{\mathrm{u}}_p$ be the balls satisfying the conclusion of Statement 11. From the definition of these balls, it follows that their boundaries $\partial B^{\mathrm{s}}_p$, $\partial B^{\mathrm{u}}_p$ are secant spheres for the restriction of the flow $f^t$ on the sets $W^{\mathrm{s}}_p\setminus p$, $W^{\mathrm{u}}_p\setminus p$, respectively. Let $g_{\mathrm{s}}\colon \partial B^{\mathrm{s}}_p\to \partial \mathbb{B}^{n-i}$, $g_{\mathrm{u}}\colon \partial B^{\mathrm{u}}_p\to \partial \mathbb{B}^{i}$ be orientation preserving homeomorphisms. We define homeomorphisms $h_{\mathrm{s}}\colon B^{\mathrm{s}}_p\to \mathbb{B}^{n-i}$, $h_{\mathrm{u}}\colon B^{\mathrm{u}}_p\to \mathbb{B}^i$ mapping trajectories of the restriction of the flow $f^t$ on the balls $B^{\mathrm{s}}_p$, $B^{\mathrm{u}}_p$ into the trajectories of the restriction of the flow $b^t$ on the balls $\mathbb{B}^{n-i}$, $\mathbb{B}^{i}$ in the following way.
To each point $x\in B^{\mathrm{s}}_p\setminus p$ we assign a time $t_x\leqslant 0$ such that $f^{t_x}(x)\subset \partial B^{ \mathrm{s}}_p$, and set
$$
\begin{equation*}
h_{\mathrm{s}}(x)=\begin{cases} b^{-t_x}(g_{\mathrm{s}}(f^{t_x}(x))), &x\in B^{\mathrm{s}}_p\setminus p, \\ O, &x=p. \end{cases}
\end{equation*}
\notag
$$
Similarly, we define a homeomorphism $h_{\mathrm{u}}\colon B^{\mathrm{u}}_p\to \mathbb{B}^i$. Namely, to each point $x\in B^{\mathrm{u}}_p\setminus p$ we assign a time $t_x\geqslant 0$ such that $f^{t_x}(x)\subset \partial B^{\mathrm{u}}_p$, and set
$$
\begin{equation*}
h_{\mathrm{u}}(x)=\begin{cases} b^{-t_x}(g_{\mathrm{s}}(f^{t_x}(x))), &x\in B^{\mathrm{u}}_p\setminus p, \\ O, &x=p. \end{cases}
\end{equation*}
\notag
$$
Each point of the set $\mathbb{B}^{n-i}\times \mathbb{B}^i$ is defined by a pair of coordinates $\xi\in \mathbb{B}^{n-i}$, $\eta\in \mathbb {B}^i$. We define the desired homeomorphism $h_p\colon V_p\to \mathbb{B}^{n-i}\times \mathbb{B}^i$ by assigning to each point $q\in V_p$ a point with coordinates $(h_{\mathrm{s}}(\pi_{\mathrm{s}}(q)),h_{\mathrm{u}}(\pi_{\mathrm{u}}(q)))$. The proposition is proved. § 4. The structure of the non-wandering set and the embedding of invariant manifolds of the flows from the class $G(M^4)$The following proposition proves the first statement of Lemma 1. Proposition 6. Let $f^t\,{\in}\, G(M^4)$ and the set $\Omega^1_{f^t}$ be non-empty. Then for an arbitrary point $p \in \Omega^1_{f^t}$ the closure $\operatorname{cl}W^{\mathrm{s}}_p$ of its stable manifold is a locally flat 3-dimensional sphere that divides the manifold $M^4$ into two connected components. Proof. Let $p \in \Omega^1_{f^t}$. By assumption, the invariant manifolds of different saddles do not intersect each other. Then, due to Corollary 3, the closure $\operatorname{cl}W^{ \mathrm{s}}_p$ of the stable manifold $W^{\mathrm{s}}_p$ of $p$ is a locally flat 3-dimensional sphere.
We show that the sphere $\operatorname{cl}W^{\mathrm{s}}_p$ divides the manifold $M^4$ into two connected components. According to Statement 1, the manifold $M^4$ is orientable. It follows from [20], Theorem 3, that a locally flat sphere $S^{n-1}$ in an orientable manifold $M^{n}$ ($n \geqslant 3$) is cylindrically embedded, so there exists a closed neighborhood $V\subset M^{n}$ of the sphere $S^{n-1}$ and a homeomorphism $h\colon \mathbb{S}^{n-1}\times [-1,1] \to V$ such that $h(\mathbb{S}^{n-1}\times \{0\})= S^{n-1}$. By definition, the neighborhood $V$ of the sphere $\operatorname{cl}W^{\mathrm{s}}_p$ is divided by this sphere into two parts. We choose points $x$, $y$ that belong to different connected components of the set $V\setminus \operatorname{cl}W^{\mathrm{s}}_p$ and join them with a smooth arc $l\subset V$ intersecting the sphere $ \operatorname{cl}W^{\mathrm{s}}_p$ at a single point. If $\operatorname{cl}W^{\mathrm{s}}_p$ does not divide $M^4$, then there exists an arc $b\subset M^4\setminus V_p$ connecting the points $x$, $y$. By construction, the intersection index of the arc $\lambda=l\cup b$ and the sphere $\operatorname{cl}W^{\mathrm{s}}_p$ is equal to $1$ or $-1$ (depending on the choice of the orientation of the arc $ \lambda$). On the other hand, since the group $\pi_{3}(M^4)$ is trivial by Proposition 1, it is easy to choose the sphere $S^{3}\subset M^4\setminus \lambda $ homotopic to the sphere $\operatorname{cl}W^{\mathrm{s}}_p$. As the intersection index is a homotopy invariant (see [24], Ch. 2, § 4), the intersection index of the sphere $S^3$ and the arc $\lambda$ must be equal to $\pm 1$, but since $S^3\cap \lambda=\varnothing$, it is zero. This contradiction proves that the sphere $\operatorname{cl}W^{\mathrm{s}}_p$ divides the manifold $M^4$ into two connected components. The proposition is proved. Remark 2. Statement 1) of Lemma 1 for a point $p \in \Omega^{3}_{f^t}$ can be obtained from Proposition 6 by considering the flow $f^{-t}$. Before we prove statement 2) of Lemma 1, we provide a range of auxiliary statements that also have independent meaning for further constructions. Recall that a set $A$ is called an attractor of a flow $f^t$ if there exists a closed neighborhood (a trapping neighborhood) $V\subset M^n$ such that all trajectories of the flow $f^t$ intersect its boundary $\partial V$ transversally, and $A=\bigcap_{t>0}f^t(V)$. A set $R$ is called a repeller of the flow $f^t$ if it is an attractor for the flow $f^{-t}$. Let $f^t\in G(M^4)$. Set
$$
\begin{equation*}
A_{f^t}=\bigcup_{p\in \Omega^0_{f^t}\cup \Omega^1_{f^t}}W^{\mathrm{u}}_{p},\qquad R_{f^t}=\bigcup_{p\in \Omega^3_{f^t}\cup \Omega^2_{f^t}\cup \Omega^4_{f^t}}W^{\mathrm{s}}_{p},\qquad M_{f^t}=M^4\setminus (A_{f^t}\cup R_{f^t}).
\end{equation*}
\notag
$$
Proposition 7. Let $f^t\in G(M^4)$. Then the set $A_{f^t}$ is a connected attractor of the flow $f^t$ with the trapping neighborhood $V_{a}$ such that: 1) $V_a$ is diffeomorphic to the ball; 2) any trajectory of the restriction of the flow $f^t$ on the set $M_{f^t}$ intersects the boundary $\Sigma$ of $V_a$ transversely and at a single point. Proof. Let $\varphi \colon M^4\to [0,4]$ be the energy function of the flow $f^t$, defined in Statement 9. Set
Since $\varphi(x)\in [0,1]$ for any point $x\in A_{f^t}$, $A_{f^t}\subset \operatorname{int} V_a$. According to Part 2) of Statement 9, trajectories of the flow $f^t$ intersect transversely the boundary of the set $V_a$ at no more than one point, and $f^t(V_a)\subset \operatorname{int} V_a$ for any $t>0$. Since the set $A_{f^t}$ is invariant, then an inclusion $A_{f^t}\subset \bigcap_{t>0}f^t(V_a)$ holds. Let us show that $A_{f^t}= \bigcap_{t>0}f^t(V_a)$, whence it will follow that $A_{f^t}$ is an attractor, $V_a$ is its trapping neighborhood, $M_{f^t}=\bigcup_{t\in \mathbb{R}}f^t(\Sigma)$, and any trajectory of the restriction of the flow $f^t$ on the set $M_{f^t}$ has a non-empty intersection with $\Sigma$. Suppose the contrary: let $A_{f^t}\neq \bigcap_{t>0}f^t(V_a)$. Then there exists a point $x\in \bigcap_{t>0}f^t(V_a)\setminus A_{f^t}$. It follows from Statement 10 that there exists an equilibrium state $p\in \Omega_{f^t}$ such that $x\in W^{\mathrm{u}}_p$. Since $V_a\subset f^{-t}(\operatorname{int} V_a)$ for any $t>0$, $\bigcap_{t>0}f^t(V_a)=\bigcap_{t\in \mathbb{R}}f^t(V_a)$. Therefore the set $\bigcap_{t>0}f^t(V_a)$ is invariant for the flow $f^t$ and the whole orbit of the point $x$ belongs to the set $\bigcap_{t>0}f^t(V_a)$. Since the set $\bigcap_{t>0}f^t(V_a)$ is closed, $p\in \bigcap_{t>0}f^t(V_a)$. However, all equilibrium states belonging to the set $\bigcap_{t>0}f^t(V_a)$ are only sinks and saddle equilibria that have Morse index equal to one, and whose unstable manifolds lie in the set $A_{f^t}$. Consequently, $p\in A_{f^t}$ and $x\in A_{f^t}$, that contradicts to the assumption. Hence $A_{f^t}= \bigcap_{t>0}f^t(V_a)$. We now prove the connectedness of $V_a$, from which the connectedness of the set $A_{f^t}$ as an intersection of connected compact nested sets follows. Suppose that $V_a$ is disconnected, that is, can be represented as a union of disjoint non-empty invariant subsets $E_1, E_2$. Then the union $\bigcup_{t\in \mathbb{R}}f^t(E_1\cup E_2)$ is also disconnected. However, according to the previous paragraph, $\bigcup_{t\in \mathbb{R}}f^t(E_1\cup E_2)=\bigcup_{p\in A_{f^t}}{W^{\mathrm{s}}_p}$. Due to Statement 10, the manifold $M^4$ can be represented as $M^4=\bigcup_{p\in A_{f^t}}{W^{\mathrm{s}}_p}\cup {R}_{f^t}$. Then $M^4\setminus R_{f^t}=\bigcup_{p\in A_{f^t}}{W^{\mathrm{s}}_p}$, therefore $M^4\setminus {R}_{f^t}$ is disconnected. On the other hand, since the dimension of the set ${R}_{f^t}$ does not exceed $2$, ${R}_{f^t}$ does not divide $M^4$, hence the set $ M^4\setminus {R}_{f^t}$ is connected. This contradiction proves that $V_a$ is connected. We claim that $A_{f^t}$ does not contain subsets homeomorphic to the circle. Assume the opposite: let $c\subset A_{f^t}$ be a simple closed curve. It follows from Part 3) of Statement 10 that the set $A_{f^t}\setminus \Omega^{1}_{f^t}$ is a finite set of frames of arcs lying in a disjointed union of the stable manifolds of the sink points. Therefore, the existence of the curve $c$ implies that there exists an equilibrium state $p\in \Omega^1_{f^t}$ belonging to this curve. By statement 1) of Lemma 1, the sphere $\operatorname{cl}W^{\mathrm{s}}_p$ divides the ambient manifold $M^4$ into two connected components. Since the one-dimensional unstable separatrices of the point $p$ are separated by the manifold $W^{\mathrm{s}}_p$, they belong to different connected components of the set $M^4\setminus \operatorname{cl}W^{\mathrm{ s}}_p$. Therefore, the curve $c$ containing $W^{\mathrm{u}}_p$, intersects the sphere $\operatorname{cl}W^{\mathrm{s}}_p$ at least at one more point $ x\in c\cap \operatorname{cl}W^{\mathrm{s}}_p$, different from $p$. The point $x$ cannot be a source, since $A_{f^t}$, by construction, does not contain sources. The point $x$ cannot be a sink point or a saddle point, since the set $W^{\mathrm{s}}_p\setminus p$ consists of wandering points. Consequently, $x$ belongs to the one-dimensional unstable manifold of some point $q\in \Omega^1_{f^t}$, that is impossible for Morse–Smale flows. Thus, the set $A_{f^t}$ can be represented as a connected graph without cycles, whose vertices are sink points, and whose edges are one-dimensional unstable manifolds of saddle points. Hence, we immediately have $|\Omega^{0}_{f^t}|=|\Omega^1_{f^t}|+1$. It follows from Morse theory (see, for example [25]) that the trapping neighborhood $V_a$ is obtained from the union of $|\Omega^0_{f^t}|$ pairwise disjoint balls by gluing $|\Omega^1_ {f^t}|$ handles of index $1$. We prove by induction on the number $k=|\Omega^{0}_{f^t}|$ that the connected compact manifold that is obtained from $k$ copies of closed $n$-balls by gluing $(k-1)$ handles of index 1, is diffeomorphic to the $n$-ball. It will mean that the set $V_a$ is diffeomorphic to the $n$-ball.  Figure 3.The union of two balls with a glued handle: (a) – the manifold is disconnected, (b) – the manifold is connected and diffeomorphic to the ball (to the proof of Proposition 7) For $k=1$ there is one ball and $0$ handles, the statement is true. Assume that the statement is proved for $i\geqslant 1$ and consider the case $k=i+1$. Then $V$ is the union of two balls with a glued handle. There are two possibilities to glue a handle, shown in Fig. 3: (a) the manifold is disconnected, which is impossible by assumption; (b) the manifold is connected and diffeomorphic to the ball. The proposition is proved. Proposition 7 and the definition of the set $R_{f^t}$ directly imply the following statement. Corollary 4. The set $R_{f^t}$ is a connected repeller of the flow $f^t$ with a trapping neighborhood $M^4\setminus V_a$. Applying Proposition 7 to the flow $f^{-t}$ proves the following fact. Corollary 5. The set is a connected attractor for the flow $f^{-t}$ for which there exists a trapping neighborhood $V_r$ that is diffeomorphic to a ball and such that $V_r\cap V_a= \varnothing$. The attractor $A_{f^t}$, repeller $\widetilde{R}_{f^t}$ and their trapping neighborhoods are shown on Fig. 4. Proposition 8. Let $\Sigma$ be the boundary of the trapping neighborhood $V_a$ of the attractor $A_{f^t}$. Then: 1) three-dimensional stable and unstable manifolds of saddle equilibria with Morse indices $1$ and $3$, respectively, intersect the sphere $\Sigma$ along smoothly embedded two-dimensional spheres; 2) the intersection $W^{\mathrm{u}}_{\sigma_2}\cap \Sigma$ of the sphere $\Sigma$ with the unstable manifold of the saddle point of the flow $f^t$ that has the Morse index $2$, is a knot. Proof. Let $\sigma_i\in \Omega^{i}_{f^t}$, $i\in \{2,3\}$. It follows from Statement 10 that the closure of the set $W^{\mathrm{u}}_{\sigma_i}$ consists of $W^{\mathrm{u}}_{\sigma_i}$ and the only sink equilibrium state. Since all the sink equilibria of the flow $f^t$ belong to the attractor $A_{f^t}$, the intersection
$$
\begin{equation*}
X^{\mathrm{u}}_{\sigma_i}=W^{\mathrm{u}}_{\sigma_i}\cap \Sigma
\end{equation*}
\notag
$$
is non-empty. Similar to the arguments in the proof of Proposition 4, where the topology of the set $X^{\mathrm{u}}_\sigma$ was studied, one can prove that the set $X^{\mathrm{u}}_{\sigma_i}$ is a smooth closed manifold of dimension $i-1$, which, moreover, is a deformation retract of the set $W^{\mathrm{u}}_{\sigma_i}\setminus \sigma_i$. It follows that for $i=2$ the set $X^{\mathrm{u}}_{\sigma_2}$ is a knot, and for $i=3$ the set $X^{\mathrm{u}}_{ \sigma_2}$ is a two-dimensional sphere.
Let $\sigma_1\in \Omega^{1}_{f^t}$. It follows from the definition of the attractor $A_{f^t}$ that $\sigma_1\subset A_{f^t}$, so the intersection $W^{\mathrm{s}}_{\sigma_1}\cap \Sigma$ is non-empty. Similarly to the previous case, we prove that the set $W^{\mathrm{s}}_{\sigma_1}\cap \Sigma$ is a two-dimensional sphere. The proposition is proved. Proof. Two cases are possible: 1) the set $\Omega^1_{f^t}\cup \Omega^3_{f^t}$ is empty; 2) at least one of the sets $\Omega^1_{f^t}, \Omega^3_{f^t}$ is non-empty.
Consider case 1). Note that it follows from Part 3) of Statement 1 that Euler characteristic of the projective-like manifold of dimension $4$ and higher equals $3$. Then for the flow $f^t$, the Poincare–Hopf formula implies that the following equality is true:
$$
\begin{equation}
\bigcup_{p\in\Omega_{f^t}}(-1)^{\operatorname{ind}_p}=3.
\end{equation}
\tag{4.1}
$$
By Corollary 2, the set $\Omega_{f^t}$ contains at least one source and one sink. Since the sources, sinks, and saddles of the flow $f^t$ whose Morse index equals 2 make a positive contribution to the left side of the formula (4.1), the condition $\Omega^1_{f^t}\,{ \cup}\, \Omega^3_{f^t}\,{=}\,\varnothing$ immediately gives that the set $\Omega_{f^t}$ consists of exactly three equilibrium states: a source, a sink, and a saddle with Morse index $2$. Consider case 2). Due to Proposition 7, the trapping neighborhood $V_a$ of the attractor
$$
\begin{equation*}
A_{f^t}=\biggl(\bigcup_{\omega\in \Omega^0_{f^t}}\omega\biggr)\cup \biggl(\bigcup_{\sigma_1\in \Omega^1_{f^t}}W^{\mathrm{u}}_{\sigma_1}\biggr)
\end{equation*}
\notag
$$
is diffeomorphic to the $n$-ball. By Corollary 5, the set
$$
\begin{equation*}
\widetilde{R}_{f^t}=\biggl(\bigcup_{\alpha\in \Omega^n_{f^t}}\alpha\biggr)\cup \biggl(\bigcup_{\sigma_{n-1}\in \Omega^{n-1}_{f^t}}W^{\mathrm{s}}_{\sigma_{n-1}}\biggr)
\end{equation*}
\notag
$$
is an attractor of the flow $f^{-t}$ and has a trapping neighborhood $V_r$ diffeomorphic to the ball, and such that $V_r\cap V_a=\varnothing$.
Remove the interiors of the balls $V_r, V_a$ from the manifold $M^4$ and glue to the resulting manifold with boundary two standard $n$-balls, on one of which the vector field $\dot{x}=x$ is given, and the field $\dot{x}=-x$, $x\in \mathbb{B}^n$, is given on the second, by means of a diffeomorphism $\varphi\colon \partial (M^4\setminus \operatorname{int}(V_r\cup V_a))\to \mathbb{B}^n\times \mathbb{S}^0$. The diffeomorphism $\varphi$ can be chosen in such a way4 that on the resulting manifold $\widetilde{M}^4$ the flow $\widetilde{f}^t\in G(M^4)$ is well defined, whose non-wandering set consists of a source, a sink, and $|\Omega^2_{f^t}|$ saddles of index $2$. The operation of gluing balls is equivalent to taking the connected sum with two spheres, so the manifold $\widetilde{M}^4$ is homeomorphic to the original manifold $M^4$. Then for the flow $\widetilde{f}^t$ we obtain the situation considered in case 1). Therefore $|\Omega^2_{\widetilde{f}^t}|=1$, hence $|\Omega^2_{{f}^t}|=1$. The proposition is proved. The proof of the following statement applies the arguments used in [9], Proposition 3, to prove the local flatness of the closures of invariant manifolds of saddle of index $2$ for a gradient-like flow $f^t\in G(M^4)$ with three states of equilibrium. Proposition 10. Let $f^t\in G(M^4)$ and $V_a$, $V_r$ be trapping neighborhoods of the attractor and repeller $A_{f^t}$, $R_{f^t}$. Then the knots are trivial. Proof. Due to Statement 11, there exists a compact neighborhood $V_{p}\subset M^4$ of $p$ and two continuous maps $\pi_{\mathrm{s}}\colon V_p\to B^ {\mathrm{s}}_p$, $\pi_{\mathrm{u}}\colon V_p\to B^{\mathrm{u}}_p$, where $B^{\mathrm{s}}_p= V_p\cap W^{\mathrm{s}}_p$, $B^{\mathrm{u}}_p=V_p\cap W^{\mathrm{u}}_p$ are two-dimensional disks containing the point $p$, endowing the neighborhood $V_p$ with the direct product structure $B^{\mathrm{s}}_p\times B^{\mathrm{u}}_p$.
The boundary $\partial V_{p}$ of the neighborhood $V_{p}$ is the $3$-sphere decomposed in a natural way into two solid tori $\Pi_{\mathrm{s}}=\partial B^{\mathrm{s}}_p\times B^{\mathrm{u}}_p$, $\Pi_{\mathrm{u}}=\partial B^{\mathrm{u}}_p\times B^{\mathrm{s }}_p$ with the common boundary $T=\partial B^{\mathrm{s}}_p\times \partial B^{\mathrm{u}}_p$. Let $x\in \partial B^{\mathrm{s}}_p$. Then the curve $\mu=\{x\}\times \partial B^{\mathrm{u}}_p$ is the meridian of the solid torus $\Pi_{\mathrm{s}}$ and the longitude of the solid torus $\Pi_{\mathrm{u}}$. It follows from Proposition 7 that for any point $x\in \Pi_{\mathrm{u}}$ there exists a trajectory $l_x$ of the flow $f^t$ intersecting $\partial V_a$ at a unique point $z_x$. Denote by $l_{x,z_x}$ the segment of the trajectory $l_x$ between the points $x$, $z_x$, and set $L_{\mathrm{u}}=\bigcup_{x\in \Pi_{\mathrm {u}}} l_{x,z_x}$. By construction, the set $L_{\mathrm{u}}$ is homeomorphic to the direct product $\mathbb{B}^2\times \mathbb{S}^1\times [0,1]$, and the intersection $\widetilde{\Pi }_{\mathrm{u}}=L_{\mathrm{u}}\cap \partial V_a$ is a tubular neighborhood of $C_{\mathrm{u}}$ in $\partial V_a$. Moreover, the correspondence $x\to z_x$ defines a homeomorphism $\chi_{\mathrm{u}}\colon \Pi_{\mathrm{u}}\to \widetilde{\Pi}_{\mathrm{u}}$, sending the core $\partial B_{\mathrm{u}}$ of the solid torus $\Pi_{\mathrm{u}}$ to the knot $C_{\mathrm{u}}$. Denote by $L_{\mathrm{s}}$, $\widetilde{\Pi}_{\mathrm{s}}\subset \partial W$, $\chi_{\mathrm{s}}\colon \Pi_{ \mathrm{s}}\to \widetilde{\Pi}_{\mathrm{s}}$ the similar objects for the solid torus $\Pi_{\mathrm{s}}$ and the sphere $\partial V_r$. It follows from Proposition 7 that for any point $w\in \partial V_r\setminus \operatorname{int} \widetilde{\Pi}_{\mathrm{s}}$ there exists a trajectory $l_w$ of the flow $ f^t$ intersecting $\partial V_a\setminus \operatorname{int} \widetilde{\Pi}_{\mathrm{u}}$ at a single point $v_w$. Denote by $\theta\colon \partial V_r\setminus \operatorname{int} \widetilde{\Pi}_{\mathrm{s}}\to \partial V_a\setminus \operatorname{int} \widetilde{\Pi}_ {\mathrm{u}}$ a homeomorphism such that $\theta(w)= v_w$. It follows from the construction that
$$
\begin{equation*}
\theta|_{\partial \widetilde{\Pi}_{\mathrm{s}}}=\chi_{\mathrm{u}} \chi_{\mathrm{s}}^{-1}|_{\partial \widetilde{\Pi}_{\mathrm{s}}},
\end{equation*}
\notag
$$
so $\theta|_{\partial \widetilde{\Pi}_{\mathrm{s}}}$ sends the meridian $\chi_{\mathrm{s}}(\mu)$ of the solid torus $\widetilde{\Pi} _{\mathrm{s}}$ to a longitude $\chi_{\mathrm{u}}(\mu)$ of the solid torus $\widetilde{\Pi}_{\mathrm{u}}$.
Thus, the sphere $\partial V_a$ can be obtained from the sphere $\partial V_r$ by removing the interior of the solid torus $\widetilde{\Pi}_{\mathrm{s}}$ and gluing the solid torus $\widetilde{\Pi}_{ \mathrm{u}}$ by means of the homeomorphism $\theta$, that is, by a non-trivial surgery along the knot $C_{\mathrm{u}}$. According to Statement 7, the manifold obtained by nontrivial surgery along a nontrivial knot is not homeomorphic to the sphere. Therefore, the knot $C_{\mathrm{u}}$ is trivial. Since the knots $C_{\mathrm{u}}$, $C_{\mathrm{s}}$ have homeomorphic complements, due to [14], Theorem 1, the knot $C_{\mathrm {s}}$ is also trivial. The proposition is proved. Applying proposition 10 to the flows $f^t$, $f^{-t}$ and using the arguments similar to those in the proof of Proposition 4, we obtain the following statement. Corollary 6. Let $f^t\in G(M^4)$, $p\in \Omega^2_{f^t}$. Then the closures of the invariant manifolds of the saddle $p$ are locally flat two-dimensional spheres. Now the statement 2) of Lemma 1 immediately follows from the Proposition 9 and Corollary 6. § 5. Necessary and sufficient conditions of topological equivalence of the flows from the class $G(M^4)$In this Section, Theorem 1 is proved. It follows directly from the definition of the topological equivalence that if flows $f^t, g^t\in G(M^4)$ are topologically equivalent, then their bi-color graphs $\Gamma_{f^t}$, $\Gamma_{g^ t}$ are isomorphic. Suppose that the graphs of the flows $f^t, g^t\in G(M^4)$ are isomorphic by means of the color-preserving isomorphism $\eta\colon \Gamma_{f^t}\to \Gamma_{g^t}$ which maps the marked vertex of the graph $\Gamma_{f^t}$ into the marked vertex of the graph $\Gamma_{g^t}$, and prove that the flows are topologically conjugated. Recall that $\mathcal{L}_{f^t}$ denotes the set of all three-dimensional spheres $\{\operatorname{cl}W^{\mathrm{s}}_p,\, p\in \Omega_{f ^t}^1\}$ and $\{\operatorname{cl}W^{\mathrm{u}}_q,\, q\in \Omega_{f^t}^{3}\}$, and $ k_{f^t}$ is the number of these spheres. By Lemma 1, each sphere from the set $\mathcal{L}_{f^t}$ divides the manifold $M^4$ into two connected components, so the set $M^4\setminus \bigl( \bigcup_{p\in \Omega_{f^t}^1}\operatorname{cl}W^{\mathrm{s}}_p \cup \bigcup_{q\in \Omega_{f^t}^{n- 1}}\operatorname{cl}W^{\mathrm{u}}_q\bigr)$ consists of $m_{f^t}=k_{f^t}+1$ connected components $D_1, \dots, D_ {m_{f^t}}$. $\mathcal{D}_{f^t}$ denotes the set of all these components. Recall that $V(\Gamma_{f^t})$ and $E(\Gamma_{f^t})$ denote the set of vertices and edges of the graph $\Gamma_{f^t}$, respectively. By the definition of the graph $\Gamma_{f^t}$, there are one-to-one correspondence
$$
\begin{equation*}
\xi^0_{f^t}\colon \mathcal D_{f^t}\to V(\Gamma_{f^t}),\qquad \xi^1_{f^t}\colon \mathcal{L} _{f^t}\to E(\Gamma)_{f^t}
\end{equation*}
\notag
$$
such that the vertices $\xi^0_{f^t}(D_i)$, $\xi^0_{f^t}(D_j)$ are incident to the edge $\xi^1_{f^t}(L)$ if and only if the domains $D_i, D_j\in \mathcal D_{f^t}$ have a common boundary component $L\in \mathcal{L}_{f^t}$. Since graphs $\Gamma_{f^t}, \Gamma_{g^t}$ are isomorphic, they have the same number of vertices and edges, hence $m_{f^t}=m_{g^ t}$. Moreover, the isomorphism $\eta\colon \Gamma_{f^t}\to \Gamma_{g^t}$ induces a one-to-one correspondence $\eta_*$ between the connected components of the sets $ \mathcal D_{f^t}\cup \mathcal{L}_{f^t}$, $\mathcal D_{g^t}\cup \mathcal{L}_{g^t}$ as follows:
$$
\begin{equation*}
\eta_*(D)={\xi^0}^{-1}_{g^t}\eta(\xi^0_{f^t}(D)), \qquad \eta_*(L)= {\xi^1}^{-1}_{g^t}\eta(\xi^1_{f^t}(L))
\end{equation*}
\notag
$$
for any $D\in \mathcal D_{f^t}$, $L\in \mathcal{L}_{f^t}$. The one-to-one correspondence $\eta_*$ can be extended to a one-to-one correspondence between the sets $\Omega_{f^t}$ and $\Omega_{g^t}$ as follows. 1. Let $\sigma\in \Omega^{1}_{f^t}$. Then there is a unique point $\alpha\subset \Omega^{n}_{f^t}$ such that $\operatorname{cl} W^{\mathrm{s}}_{\sigma}=W^{\mathrm{s}}_{\sigma}\cup \alpha$. Moreover, there is a unique pair of points $\sigma'\in \Omega^{1}_{g^t}$, $\alpha'\in \Omega^n_{g^t}$ such that $\operatorname{ cl} W^{\mathrm{s}}_{\sigma'}=W^{\mathrm{s}}_{\sigma'}\cup \alpha'$ and $\eta_{*}(\operatorname{ cl} W^{\mathrm{s}}_{\sigma})=\operatorname{cl} W^{\mathrm{s}}_{\sigma'}$. Set $\eta(\sigma)=\sigma'$, $\eta(\alpha)=\alpha'$. 2. Let $\sigma\in \Omega^{n-1}_{f^t}$. Then there is a unique point $\omega\subset \Omega^{0}_{f^t}$ such that $\operatorname{cl} W^{\mathrm{u}}_{\sigma}=W^{\mathrm{u}}_{\sigma}\cup \,\omega$, and a unique pair of points $\sigma'\in \Omega^{n-1}_{g^t}$, $\omega'\in \Omega ^0_{g^t}$ such that $\operatorname{cl} W^{\mathrm{u}}_{\sigma'}=W^{\mathrm{u}}_{\sigma'}\cup \, \omega'$ and $\eta_{*}(\operatorname{cl} W^{\mathrm{u}}_{\sigma})=\operatorname{cl} W^{\mathrm{u}}_{\sigma'}$. We set $\eta(\sigma)=\sigma'$, $\eta(\omega)=\omega'$. 3. Let $\omega\in \Omega^0_{f^t}$ ($\alpha\in \Omega^n_{f^t})$ be such a point that $W^{\mathrm{s}}_{\omega}$ ($W^{\mathrm{u}}_{\alpha}$) does not intersect any three-dimensional separatrix of saddle equilibria of the flow $f^t$. Then $\omega$ ($\alpha$) belongs to the unique domain $D\in \mathcal{D}_{f^t}$ containing the closures of a one-dimensional separatrices of those saddle equilibria whose three-dimensional invariant manifolds form the boundary of the domain $D$. Moreover, in the domain $\eta_*(D)$ there is a unique sink (source) point $\omega'$ ($\alpha'$) of the flow $g^t$, which also does not intersect with any one three-dimensional separatrix of saddle equilibrium states of the flow $g^t$. We set $\omega'=\eta_*(\omega)$ ($\alpha'=\eta_*(\alpha)$). Everywhere below, for an arbitrary equilibrium state $p\in \Omega_{f^t}$ we denote by $p'$ an equilibrium state from the set $\Omega_{g^t}$ such that $p'=\eta_{*}(p)$. Recall that in Proposition 7, the section $\Sigma$ of the restriction of the flow $f^t$ to the set $M_{f^t}=M^n\setminus (A_{f^t}\cup R_{f^t})$ is defined. Denote by $\Sigma'$ the similar section for the flow $g^t$. The construction of the homeomorphism $H\colon M^4\to M^4$, which maps the trajectories of the flow $f^t$ into the trajectories of the flow $g^t$, will be described in steps, each of which will be formulated as a separate proposition. Proposition 11 (Step 1). Let $\sigma_i\in \Omega^i_{f^t}$, $\sigma'_i\in \Omega^i_{g^t}$, $i\in \{1,2,3\}$ be saddle equilibria and $V_{\sigma_i}$, $V_{\sigma'_i}$ be their canonical neighborhoods defined in Statement 11. Then there is a homeomorphism such that for all $t\in \mathbb{R}$ for which the right and left sides of the equality are defined. Proof. By Proposition 5, there is a homeomorphism $h_{\sigma_i}\colon V_{\sigma_i}\to \mathbb{B}^{n-i}\times \mathbb{B}^i$, conjugating the flow $f^t|_{V_{\sigma_i}}$ with the linear flow $b^t_i(x,y)=(e^{-t}x,e^ty)$, $x\in \mathbb {R}^{n-i}$, $y\in \mathbb{R}^{i}$.
We define the desired homeomorphism by $h_{\sigma_i,\sigma'_i}=h_{\sigma'_i}^{-1}h_{\sigma_i}$. The proposition is proved. Recall that, according to Statement 11 and Proposition 5, the canonical neighborhood $V_{\sigma_i}$ of any point $\sigma_i\in \Omega^i_{f^t}$ can be represented as a direct product of two balls $B^{\mathrm{s}}_{\sigma_i}\subset W^{\mathrm{s}}_{\sigma_i}$, $B^{\mathrm{u}}_{\sigma_i }\subset W^{\mathrm{u}}_{\sigma_i}$ of dimensions $(n-i)$, $i$, respectively. The boundary of the neighborhood $V_{\sigma_i}$ is represented as the union of two sets with a common boundary as follows:
$$
\begin{equation*}
\partial V_{\sigma_i}=\partial B^{\mathrm{s}}_{\sigma_i}\times B^{\mathrm{u}}_{\sigma_i}\cup B^{\mathrm{s} }_{\sigma_i}\times \partial B^{\mathrm{u}}_{\sigma_i}.
\end{equation*}
\notag
$$
In the case $i=1$ ($i=3$) the set $\partial B^{\mathrm{s}}_{\sigma_i}\times B^{\mathrm{u}}_{\sigma_i }$ ($B^{\mathrm{s}}_{\sigma_i}\times \partial B^{\mathrm{u}}_{\sigma_i}$) is homeomorphic to the annulus $\mathbb{S}^2 \times [0,1]$, and the set $B^{\mathrm{s}}_{\sigma_i}\times \partial B^{\mathrm{u}}_{\sigma_i}$ ($\partial B^{\mathrm{s}}_{\sigma_i}\times B^{\mathrm{u}}_{\sigma_i}$) is the union of two 3-dimensional balls. In the case $i=2$ both sets $\partial B^{\mathrm{s}}_{\sigma_i}\times B^{\mathrm{u}}_{\sigma_i}$, $B^{\mathrm{s}}_{\sigma_i}\times \partial B^{\mathrm{u}}_{\sigma_i}$ are solid tori. Proposition 12 (Step 2). The spheres $\Sigma$, $\Sigma'$ can be modified in such a way that the resulting locally flat spheres (which we denote by the same symbols) have the following properties: 1) the spheres $\Sigma$, $\Sigma'$ are global sections (in the topological sense) for the flows $f^t|_{M_{f^t}}$, $g^t|_{M_{ g^t}}$, i. e. any trajectory $l_x\in M_{f^t}$ ($l_{x'}\in M_{g^t}$) intersects $ \Sigma$ ($\Sigma'$) at a single point; 2) for any saddle points $\sigma_i\in \Omega^i_{f^t}$, $\sigma_i'\in \Omega^i_{g^t}$, the intersections $V_{\sigma_i}\cap \Sigma $, $V_{\sigma'_i}\cap \Sigma'$ are non-empty and consist of sets
Proof. Let $i=1$. Set $\Pi=\partial B^{\mathrm{s}}_{\sigma_1}\times B^{\mathrm{u}}_{\sigma_1}$. Suppose that $V_{\sigma_1}\cap \Sigma\neq \Pi$. Join the set $\Pi$ to the sphere $\Sigma$ by segments of the trajectories of the flow $f^t$ and denote by $\widetilde{\Pi}$ the locus of the ends of these trajectories, so that $\widetilde{\Pi}\subset \Sigma$. By definition, $\widetilde{\Pi}$ is a submanifold of $\Sigma$, so there is an embedding $e\colon \partial \widetilde{\Pi}\times [0,1]\to \Sigma$ such that $e(\partial \widetilde{\Pi}\times [0,1])\cap \widetilde{\Pi}=e(\partial \widetilde{\Pi}\times \{0\} )$ and $e(\partial \widetilde{\Pi}\times [0,1])\cap V_{\sigma}=\varnothing$ for any saddle equilibrium different from $\sigma_1$. On the set $\widetilde{\Pi}$, a continuous function $T\colon \widetilde{\Pi}\to \mathbb{R}$ is defined which assigns a time $t_x$ such that $f^{t_x}(x)\subset \Pi$ to each point $x\in \widetilde{\Pi}$. We extend this function continuously to the set $K= e(\partial \widetilde{\Pi}\times [0,1])$ so that $T(x)=0$ for all points $x\in e(\partial \widetilde{\Pi}\times \{1\})$. As a new section, take the following sphere:
We do a similar procedure for all the remaining saddle points, as a result we get a section with the required properties, which we again denote by $\Sigma$. We modify the sphere $\Sigma'$ in the same way. The proposition is proved. For the equilibrium state $\sigma_1\in \Omega^1_{f^t}$ we set $S^{\mathrm{s}}_{\sigma_1}=W^{\mathrm{s}}_{\sigma_1}\cap \Sigma$, and for the equilibrium state $\sigma_i\in \Omega^i_{f^t}$, $i\in \{2,3\}$, we set $S^{\mathrm{u}}_{ \sigma_i}=W^{\mathrm{u}}_{\sigma_i}\cap \Sigma$. We use similar notation for saddle equilibria $\sigma'_i$, $i\in \{1,2,3\}$, of the flow $g^t$. Proposition 13 (Step 3). There is a homeomorphism $\Phi\colon \Sigma\to \Sigma'$ such that 1) $\Phi(S^{\mathrm{s}}_{\sigma_1})=S^{\mathrm{s}}_{\sigma'_1}$ for any pair of saddle points $\sigma_1\in \Omega^1_{f^t}$, $\sigma'=\eta_{*}(\sigma)\in \Omega^1_{g^t}$; 2) $\Phi(S^{\mathrm{u}}_{\sigma_i})=S^{\mathrm{u}}_{\sigma'_i}$ for any pair of saddle points $\sigma_i\in \Omega^i_{f^t}$, $\sigma'_i=\eta_{*}(\sigma_i)\in \Omega^i_{g^t}$, $i\in \{2,3\} $. Proof. Let $\sigma_2\in \Omega^2_{f^t}$, $\sigma'_2\in \Omega^2_{g^t}$. Then $\sigma'_2=\eta_*(\sigma_2)$. By Propositions 8, 10, the sets $S^{\mathrm{u}}_{\sigma_2}$, $S^{\mathrm{u}}_{\sigma'_2 }$ are trivial knots. It follows from the definition of a trivial knot that there exists a homeomorphism such that In what follows, we denote the images of the sets $S^{\mathrm{s}}_{\sigma_1}$, $S^{\mathrm{u}}_{\sigma_3}$ (for all $\sigma_1\in \Omega^1_ {f^t}$, $\sigma_3\in \Omega^3_{f^t}$) with respect to the homeomorphism $\Psi$ with the same symbols as the originals.
Let $D'\subset \Sigma'$ be a 3-ball disjoint from $\bigcup_{\sigma\in \Omega^1_{f^t}\cup \Omega^3_{f^t} }V_\sigma$ and such that $S^{\mathrm{u}}_{\sigma'_2}\subset \operatorname{int} D'$. Set $D=\Sigma'\setminus \operatorname{int} D'$. Proposition 8 implies that for any point $\sigma_1\in \Omega^1_{f^t}$ ($\sigma_3 \in \Omega^3_{f^t}$) the set $ S^{\mathrm{s}}_{\sigma_1}$ ($S^{\mathrm{u}}_{\sigma_3}$) is a smoothly embedded 2-sphere. It follows from the Generalized Schoenflies Theorem that the sphere $S^{\mathrm{s}}_{\sigma_1}$ ($S^{\mathrm{u}}_{\sigma_3}$) divides the 3-sphere $\Sigma'$ into two connected components, the closure of each is a 3-ball. Denote by $D_{\sigma_i}$, $i\in \{1,3\}$, the ball that belongs to the interior of the ball $D$. We enumerate the saddle points in such a way that for some $n_0\leqslant m_{f^t}$ the equality
$$
\begin{equation*}
\bigcup_{i=1}^{n_0}D_{\sigma_i}=\bigcup_{i=1}^{m_{f^t}}D_{\sigma_i}
\end{equation*}
\notag
$$
holds.
We use similar notation for the saddle points of the flow $g^t$. Since the bi-color graphs of the flows $f^t$, $g^t$ are isomorphic, the numbering on the set of saddle points of the flow $g^t$ can be chosen in a similar way, with $\sigma'_i=\eta_*(\sigma_i) $ for any $i\in \{1,\dots,m_{f^t}\}$. It follows from Statement 8 that there exists a homeomorphism such that1) $\Phi_0|_{D'}=\operatorname{id}$; 2) $\Phi_0(D_{\sigma_i})=D_{\sigma'_i}$, $i\in \{1,\dots,n_0\}$. If $n_0=m_{f^t}$, then we set $\Phi=\Phi_0\Psi$ and go to the next step. Suppose that $n_0<m_{f^t}$. We denote the images of the balls $D_{\sigma_i}$, $D_{\sigma'_i}$, $i\in \{1,\dots,m_{f^t}\}$, and their boundaries with respect to the homeomorphism $\Phi_0$ by the same characters as the originals. For each ball $D_{\sigma_j}$, $j\in \{1,\dots,n_0\}$, having non-empty intersection with $\bigcup_{i=n_0+1}^{m_f}D_{\sigma_i}$, we denote by $D_{\sigma_{j,1}},\dots,D_{\sigma_{j,n_j}}$ the pair-wise disjoint discs that are elements of the set $D_{\sigma_j}\cap \bigcup_{i=n_0+1}^{m_f}D_{\sigma_i}$ and such that $\bigcup_{k=1}^{n_j}D_{\sigma_{j,k}}=D_{\sigma_j}\cap \bigcup_{i=n_1+1}^{m_f}D_{\sigma_i}$, put $D_{\sigma'_{j,k}}=\eta_*(D_{\sigma'_{j,k}})$, $k\in \{1,\dots,n_j\}$, and construct a homeomorphism $\Phi_j\colon \Sigma'\to \Sigma'$ identical outside the disk $D_{\sigma_{j}}$ and such that $\Phi_j(D_{\sigma_{j,k}})= D_{\sigma'_{j,k}}$. If $D_{\sigma_j}\cap \bigcup_{i=n_0+1}^{m_f} D_{\sigma_i}= \bigcup_{ l=1}^{k_j} D_{\sigma_{j,l}}$, then the required homeomorphism $\Phi$ is a superposition of the homeomorphisms $\Psi$, $\Phi_0$ and the constructed homeomorphisms $\Phi_1,\dots, \Phi_{n_0}$. Otherwise, we continue the process and in a finite number of steps we obtain $\Phi$ as a superposition of all the constructed homeomorphisms. The proposition is proved. Proposition 14 (Step 4). There exists a homeomorphism $H_{\Sigma,\Sigma'}\colon \Sigma\to \Sigma'$ such that $H_{\Sigma,\Sigma'}|_{V_{\sigma}}=h_{\sigma, \sigma'}|_{V_{\sigma}}$ for any saddle equilibria $\sigma\in \Omega_{f^t}, \sigma'\in \Omega_{g^t}$. Proof. Let $\Phi\colon \Sigma\to \Sigma'$ be the homeomorphism constructed at Step 3. Set $\Pi_{\sigma}=\Phi(V_{\sigma}\cap \Sigma)$ for any saddle equilibrium $\sigma\in \Omega_{f^t}$, $\Pi_{\sigma'}=V_{\sigma'}\cap \Sigma'$ for a saddle $\sigma'\in \Omega_{g^t}$, and define a homeomorphism by
Let $\sigma_1\in \Omega^1_{f^t}$, $\sigma'_1=\eta_{*}(\sigma_1)\in \Omega^1_{g^t}$. The sets $\Pi_{\sigma_1}$, $\Pi_{\sigma'_1}$ are homeomorphic to the pair of balls $\mathbb{B}^3\times \mathbb{S}^0$. Let $\Pi^0_{\sigma'_1}\subset \Sigma'$ be a pair of 3-ball such that 1) $\Pi_{\sigma_1}, \Pi_{\sigma'_1}\subset \operatorname{int}\Pi^0_{\sigma'_1}$; 2) $\Pi^0_{\sigma'_1}\cap \Pi_{\sigma}=\varnothing$, $\Pi^0_{\sigma'_1}\cap \Pi_{\sigma'}=\varnothing$ for any saddle equilibria $\sigma$, $\sigma'$ of the flows $f^t$, $g^t$, respectively, different from $\sigma_1$, $\sigma'_1$. According to Statement 8 there exists a homeomorphism with the following properties:
$$
\begin{equation*}
\Psi_1|_{\Sigma'\setminus \operatorname{int} \Pi^0_{\sigma_1}}=\operatorname{id},\qquad \Psi_1|_{\Pi_{\sigma_1}}=h'_{\sigma_1,\sigma'_1}|_{\Pi'}.
\end{equation*}
\notag
$$
We construct a similar homeomorphism for each saddle point of index $1$ and denote by $\mathbf{\Psi}_1$ the superposition of the constructed homeomorphisms. Let $\sigma_2\in \Omega^2_{f^t}$, $\sigma'_2=\eta_{*}(\sigma_2)$, $S^{\mathrm{u}}_{\sigma'_2}=W^{\mathrm{u}}_{\sigma'_2}\cap \Sigma'$. In this case the sets $ \Pi_{\sigma_2}$, $\Pi_{\sigma'_2}$ are solid tori. Let $\Pi^0_{\sigma'_2}\subset \Sigma'$ be a tubular neighborhood of the knot $S^{\mathrm{u}}_{\sigma'_2}$ such that $\Pi_{\sigma_2}, \Pi'_{\sigma'_2}\subset \operatorname{int} \Pi^0_{\sigma'_2}$, $\Pi^0_{\sigma'_2}\cap \Pi_{\sigma}=\varnothing$, $\Pi^0_{\sigma'_2}\cap \Pi_{\sigma'}=\varnothing$ for any saddles $\sigma$, $\sigma'$ different from $\sigma_2$, $\sigma'_2$. By definition, the homeomorphism $h'|_{\sigma,\sigma'}$ maps the longitude and the meridian of the solid torus $\Pi_{\sigma_2}$ into the longitude and the meridian of the solid torus $\Pi'_{\sigma'_2}$, correspondingly, hence, by Proposition 2, there exists a homeomorphism such that
$$
\begin{equation*}
\Psi_2|_{\Sigma'\setminus \operatorname{int} \Pi^0_{\sigma'_2}}=\operatorname{id},\qquad \Psi_2|_{\Pi_{\sigma_2}}=h'_{\sigma_2,\sigma'_2}|_{\Pi_{\sigma_2}}.
\end{equation*}
\notag
$$
Let $\sigma_3\in \Omega^3_{f^t}$, $\sigma'_3=\eta_{*}(\sigma)\in \Omega^3_{g^t}$. The sets $\Pi_{\sigma_3}$, $\Pi_{\sigma'_3}$ are homeomorphic to the direct product $\mathbb{S}^2\times[0,1]$. Let $\Pi^0_{\sigma'_3}$ be a tubular neighborhood of the set $S^{\mathrm{s}}_{\sigma'_3}=W^{\mathrm{u}}_{\sigma'_3}\cap \Sigma'$ with properties similar to ones of the sets $\Pi^0_{\sigma'_1}$, $\Pi^0_{\sigma'_2}$. According to Proposition 3, there exists a homeomorphism such that
$$
\begin{equation*}
\Psi_3|_{\Sigma'\setminus \operatorname{int} \Pi^0_{\sigma'_3}}=\operatorname{id},\qquad \Psi_3|_{\Pi_{\sigma_3}}=h'_{\sigma_3,\sigma'_3}|_{\Pi_{\sigma_3}}.
\end{equation*}
\notag
$$
We construct a similar homeomorphism for each saddle point of index $3$ and denote by $\mathbf{\Psi}_3$ the superposition of the constructed homeomorphisms. Now the desired homeomorphism $H_{\Sigma,\Sigma'}$ is defined as a superposition $\mathbf{\Psi}_3 \Psi_2 \mathbf{\Psi}_1\Phi$. The proposition is proved. Proposition 15 (Step 5). The homeomorphism $H_{\Sigma,\Sigma'}\colon \Sigma\to \Sigma'$ can be extended to a homeomorphism $H\colon M^4\to M^4$ such that $Hf^t=g^tH$. Proof. For any saddle point $\sigma (\sigma')$ of the flow $f^t (g^t)$ set
$$
\begin{equation*}
\mathbb{V}_{\sigma}=\bigcup_{t\in \mathbb{R}} f^t(V_\sigma),\qquad \mathbb{V}_{\sigma'}=\bigcup_{t\in \mathbb{R}} g^t(V_\sigma)
\end{equation*}
\notag
$$
and define a homeomorphism by where $t_x\in \mathbb{R}$ is the time such that $f^{t_x}(x)\in V_{\sigma}$.
For any point $x\in M_{f^t}$ set
$$
\begin{equation*}
H_{M_{f^t}, M_{g^t}}(x)=g^{-t_x}\bigl(H_{\Sigma,\Sigma'}(f^{t_x}(x))\bigr),
\end{equation*}
\notag
$$
where $t_x\in \mathbb{R}$ satisfies $f^{t_x}(x)\in \Sigma$.
By construction, the homeomorphism $H_{M_{f^t},M_{g^t}}$ coincides with the homeomorphism $H_{\sigma,\sigma'}$ on the intersection $M_{f^t}\cap \mathbb{V}_{\sigma}$, so the following formula defines a homeomorphism which can be unequivocally extended to the set $\Omega^0_{f^t}\cup \Omega^4_{f^t}$: The proposition is proved.§ 6. Realization of the topological equivalence classes of the flows from the class $G(M^4)$ Proof. Each edge $e$ of the graph $\Gamma_{f^t}$ corresponds to an $(n-1)$-dimensional sphere which, by virtue of Lemma 1, divides the supporting manifold $M^4$ into two connected components. Therefore, the edge $e$ divides the graph $\Gamma_{f^t}$ into two connected components, hence the graph $\Gamma_{f^t}$ does not contain cycles.
We now show that the graph $\Gamma_{f^t}$ is connected. By definition, the graph $\Gamma_{f^t}$ has $k_{f^t}$ edges and $k_{f^t}+1$ vertices. It is known that a connected graph with $k+1$ vertices is a tree if and only if it has exactly $k$ edges. Assume that the graph $\Gamma_f$ is not connected. Then it consists of connected subgraphs $\Gamma_1,\dots, \Gamma_m$, $m\geqslant 2$, and adding $m-1$ edges turns the set of subgraphs $\Gamma_i$ into a connected graph without cycles (i. e. the tree) with $k_f+1$ vertices and $k_f+m$ edges, that contradicts the mentioned property of trees. Therefore, the graph $\Gamma_{f^t}$ is connected and does not contain cycles, that is, it is a tree. The proposition is proved. Proof of Theorem 2. Let $\Gamma$ be an arbitrary tree with one marked vertex, whose edges are colored in two colors $\mathrm s$ and $\mathrm u$. Without loss of generality, suppose that the marked vertex is incident to at least one edge of color $\mathrm s$. In the opposite case, we reverse the colors of the edges (replacing $\mathrm s$ with $\mathrm u$ and vice versa) and construct the flow $f^t$ with the resulting graph. The desired flow will be the flow $f^{-t}$.
Denote by $g_1^t\in G(M^4)$ the flow whose set of saddle equilibria consists of exactly one saddle (then its Morse index equals $2$). The algorithm of constructing such a flow is described in [9], [10]. Let $\omega\in \Omega^{0}_{g_1^t}$, $\sigma_*\in \Omega^2_{g_1^t}$. Denote by $g_2^t$ a gradient-like flow on the sphere $S^4$ whose bi-color graph $\Gamma_{g_2^t}$ is isomorphic to the graph $\Gamma$. The algorithm of constructing such flows is described in paper [27]. Let $D\in S^4$ be the domain corresponding to the marked vertex of the graph $\Gamma$. By assumption, the boundary of the domain includes at least one stable manifold of dimension $n-1$ of the saddle point $\sigma\,{\in}\, \Omega^{1}_{g_2^t}$. Consequently, the boundary of $D$ also includes the source equilibrium state $\alpha$, which belongs to the closure of the manifold $W^{\mathrm{s}}_{\sigma}$. Denote by $S_{\omega}^{n-1}\subset W^{\mathrm{s}}_{\omega}$, $S_{\alpha}^{n-1}\subset W^{\mathrm{u}}_{\alpha}$ secant spheres to the restriction of the flows $g_1^t$, $g_2^t$ to the sets $W^{\mathrm{s}}_{\omega}\setminus \omega$ , $W^{\mathrm{u}}_{\alpha}\setminus \alpha$, respectively (such spheres can be chosen as level hypersurfaces of the Morse energy functions for the flows $g_1^t$, $g_2^t$) and by $B^n_{\omega}, B^n_\alpha$ the balls bounded by the spheres $S_{\omega}^{n-1}$, $S_{\alpha}^{n-1}$ such that $\omega\in B^n_{\omega}$, $\alpha\in B^n_\alpha$. Choose the orientation of the spheres $S_{\omega}^{n-1}$, $S_{\alpha}^{n-1}$ as of the boundaries of the balls $B^n_{\omega}$, $B^n_\alpha$. Denote by $\varphi\colon S_{\omega}^{n-1}\to S_{\alpha}^{n-1}$ a diffeomorphism reversing the chosen orientation and such that $\varphi(S_{\omega}^{n-1}\cap W^{\mathrm{s}}_{\sigma_*})\,{\subset}\, D\cap S_{\alpha}^{n-1}$, glue the manifolds $M^4\setminus \operatorname{int} B^{n}_\omega$, $S^4\setminus \operatorname{int} B^n_{\alpha}$ by means $\varphi$, denote by $\widetilde{M}^4$ the obtained manifold and by $p\colon (M^4\setminus \operatorname{int} B^{n}_\omega)\cup S^4\setminus \operatorname{int} B^n_{\alpha}$ the canonical projection. The manifold $\widetilde{M}^4$ is the connected sum of the complex projective plane $M^4$ and the sphere $S^4$, so it is homeomorphic to $M^4$. A slight modification of the flows $g_1^t$, $g_2^t$ in the neighborhood of the spheres $S_{\omega}^{n-1}$, $S_{\alpha}^{n-1}$ (see details in [26]) allows us to define a flow $f^t\in G(\widetilde{M}^4)$ on the manifold $\widetilde{M}^4$ such that $f^t$ coincides with the flow $g_1^t$ on the set $p(M^4\setminus \operatorname{int} B^{n}_\omega)$ and with the flow $g_2^t$ on the set $S^4\setminus \operatorname{int} B^n_{\alpha}$. By construction, the graph $\Gamma_{f^t}$ is isomorphic to the graph $\Gamma$. Theorem 2 is proved. 
Citation in format AMSBIB
\Bibitem{GriGur22} |
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