| Russian Mathematical Surveys |
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This article is cited in 2 scientific papers (total in 2 papers) Brief Communications Topological classification of flows without heteroclinic intersections on a connected sum of manifolds V. Z. Grines, E. Ya. Gurevich National Research University "Higher School of Economics" (Nizhny Novgorod branch)
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     Let $f^t$ be a smooth flow on a closed manifold $M^n$, let the non-wandering set $\operatorname{NW}(f^t)$ of $f^t$ consist of a finite number of hyperbolic equilibrium states whose unstable manifolds have dimension (Morse index) in the range $\{0,1,n-1,n\}$, and assume that the invariant manifolds of different saddle equilibria do not intersect each other. We say that such a flow $f^t$ has no heteroclinic trajectories. According to [4], $M^n$ is homeomorphic to the manifold $\mathcal{S}^n_g$, where $\mathcal{S}^n_g$ is either the sphere $\mathbb{S}^n=\{x\in \mathbb{R}^n\colon |x|=1\}$ (for $g=0$) or the connected sum of $g>0$ copies of $\mathbb{S}^{n-1}\times \mathbb{S}^{1}$. Throughout, by an $n$-dimensional sphere $S^n$ we mean a manifold homeomorphic to $\mathbb{S}^n$. Denote by $G(\mathcal{S}^n_g)$ the class of smooth flows without heteroclinic trajectories on $\mathcal{S}^n_g$ whose non-wandering set consists of hyperbolic equilibrium states. For such flows the following statement holds. Statement 1. Let $f^t\in G(\mathcal S^n_g)$, where $g\geqslant 0$ and $n\geqslant4$. Then the Morse index of any saddle equilibrium of the flow $f^t$ equals either $1$ or $n-1$. For $g=0$ Statement 1 was proved in [8]. Suppose that $g>0$ and a flow $f^t\in G(\mathcal{S}^n_g)$ has a saddle equilibrium $\sigma$ whose Morse index $i$ belongs to the set $\{2,\dots,n-2\}$. Then by [10], Theorem 2.1, there exists a unique pair $\alpha$, $\omega$ of a source and a sink equilibrium such that $\operatorname{clos}W^{\rm s}_{\sigma}=W^{\rm s}_{\sigma}\cup \alpha$ and $\operatorname{clos}W^{\rm u}_{\sigma}=W^{\rm u}_{\sigma}\cup \omega$. Hence the closures of the stable and unstable invariant manifolds of $\sigma$ are spheres of dimension $n-i$ and $i$, respectively, which are smoothly embedded at all points except possibly at $\alpha$ and $\omega$. These spheres have the unique common point $\sigma$, and therefore their intersection number equals one in absolute value. It was shown in [3] that the homology groups $H_{i}(\mathcal{S}^n_g)$ and $H_{n-i}(\mathcal{S}^n_g)$ are trivial. This implies that there is a sphere $S^{i}$ that is homologous to the sphere $\operatorname{clos}{W^{\rm u}_{\sigma}}$ and disjoint from the sphere $\operatorname{clos}{W^{\rm s}_{\sigma}}$. Then the intersection number of $S^{i}$ and $\operatorname{clos}W^{\rm s}_{\sigma}$ is zero. It follows from [9], § 70, Theorem I, that the intersection numbers of the spheres $S^{i}$ and $\operatorname{clos}W^{\rm s}_{\sigma}$ and of $\operatorname{clos}W^{\rm u}_{\sigma}$ and $\operatorname{clos}W^{\rm s}_{\sigma}$ must be equal, so we obtain a contradiction. It follows from [2], [7], and [6] that all flows in the class $G(\mathcal{S}^n_g)$ are structurally stable. A problem of the topological classification of structurally stable flows with finite non-wandering set on manifolds has a long and rich history, beginning in the classical works [2] and [1]. However, the most complete results have been obtained for dimension $n\leqslant 3$, while for $n>3$ there are only few results (see the survey [5]). A topological classification of flows in the class $G(S^n)$, $n\geqslant 3$, was obtained in [8], where it was proved that one complete topological invariant of such flows is the phase diagram, a combinatorial invariant which generalizes the Leontovich–Mayer diagram of a dynamical system and the Peixoto graph used for the topological classification of structurally stable flows on two-dimensional manifolds. For $n=3$ the topological classification of flows in the class $G(\mathcal{S}^n_g)$ in combinatorial terms follows from more general results of [11]. In [12] an example of a Morse–Smale flow on a 4-manifold was constructed in which the closures of two-dimensional invariant manifolds of saddle equilibria are wild spheres. This example shows the fundamental impossibility of classifying multidimensional flows in combinatorial terms. Statement 1 allows one to extend significantly the class of manifolds whose partition into the trajectories of the flow still admits a combinatorial description. Let $\mathcal{L}_{f^t}$ be the set of closures of the $(n-1)$-dimensional invariant manifolds of saddle equilibrium states. By virtue of Statement 1, each element of this set is a sphere of dimension $n-1$. Let $\mathcal{D}_{f^t}$ be the set of connected components of the manifold obtained from $M^n$ by removing all spheres belonging to $\mathcal{L}_{f^t}$. The bi-colour graph of the flow $f^t$ is the graph $\Gamma_{f^t}$ with the following properties: 1) the vertex set $V(\Gamma_{f^t})$ of $\Gamma_{f^t}$ is in a one-to-one correspondence with the set $\mathcal{D}_{f^t}$, and the edge set $E(\Gamma_{f^t})$ of $\Gamma_{f^t}$ is in a one-to-one correspondence with the set $\mathcal{L}_{f^t}$; 2) two vertices $v_i$ and $v_j$ are incident to an edge $e_{i,j}$ if and only if the corresponding domains $D_i,D_j\in \mathcal{D}_{f^t}$ have a common boundary component; 3) an edge $e_{i,j}$ has colour $\mathrm{s}$ (colour $\mathrm{u}$) if it corresponds to a manifold $\operatorname{clos}W^{\rm s}_p\in \mathcal{L}_{f^t}$ (to $\operatorname{clos}W^{\rm u}_q \in \mathcal{L}_{f^t}$, respectively). Theorem 1. Two flows $f^t,f'^t\in G(\mathcal{S}^n_g)$ are topologically equivalent if and only if their graphs $\Gamma_{f^t}$ and $\Gamma_{f'^t}$ are isomorphic by means of a colour-preserving isomorphism. The idea of the proof of Theorem 1 is as follows. Let $\Omega^i_{f^t}$ denote the set of all equilibrium states of the flow $f^t\in G(\mathcal{S}^n_g)$ whose unstable manifolds have dimension $i\in \{0,1,n-1,n\}$. Set
$$
\begin{equation*}
A_{f^t}=\Omega^0_{f^t}\cup W^{\rm u}_{\Omega^1_{f^t}},\quad R_{f^t}=\Omega^n_{f^t}\cup W^{\rm s}_{\Omega^{n-1}_{f^t}},\quad\text{and}\quad V_{f^t}=\mathcal{S}^n_g\setminus (A_{f^t}\cup R_{f^t}).
\end{equation*}
\notag
$$
We show that the restriction of $f^t$ to $V_{f^t}$ has a global section $\Sigma_{f^t}$, and the existence of an isomorphism between the graphs $\Gamma_{f^t}$ and $\Gamma_{{f'}^t}$ leads to the existence of a homeomorphism $h\colon \Sigma_{f^t}\to \Sigma_{{f'}^t}$ sending the intersection of $\Sigma_{f^t}$ with the invariant manifolds of saddle equilibria of $f^t$ to the intersection of $\Sigma_{{f'}^t}$ with the invariant manifolds of saddle equilibria of the flow ${f'}^t$. Then we extend $h$ to the sets $V_{f^t}$, $A_{f^t}$, and $R_{f^t}$, to a homeomorphism mapping all the trajectories of $f^t$ to trajectories of ${f'}^t$.
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\Bibitem{GriGur22} |
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