On gradient-like flows on manifolds of dimension four and greater

We will say that gradient-like flow $f^t$ belongs to a class $G(M^n)$, where $M^n$ is connected closed oriented manifold of dimension $n\geq 3$, if:

• Morse index (dimension of unstable manifold) of any saddle equilibrium state of the flow $f^t$ equals either $1$ or $n-1$;
• invariant manifolds of different saddle equilibria do not intersect.

Denote by $\nu_{f^t}$ and $\mu_{f^t}$ the numbers of saddle and node equilibria of the flow $f^t\in G(M^n)$ and set
$$ g_{f^t}=(\nu_{f^t}-\mu_{f^t}+2)/2. $$

Everywhere below $\mathcal{S}^n_g$ stands for manifold which homeomorphic either to the sphere $\mathbb{S}^n$ if $g=0$ or to connected sum of $g>0$ copies of $\mathbb{S}^{n-1}\times \mathbb{S}^{1}$.
Theorem 1. Let $f^t\in G(M^n)$, $n\geq 2$. Then $M^n$ is homeomorphic to $\mathcal{S}^n_{g_{f^t}}$.
For $n=2$ Theorem 1 immediately follows from [5]. For $n\geq 3$ Theorem 1 follows from [1, 2], where its analog for Morse-Smale diffeomorphisms without heteroclinic curves was obtained.
Theorem 2 below states that for manifolds $\mathcal{S}^n_g$, $n\geq4$, the condition (b) implies the condition (a).
Theorem 2. Let $f^t$ be gradient-like flow on $\mathcal{S}^n_g$, $g\geq 0$, $n\geq4$. If invariant manifolds of different saddle equilibria of $f^t$ do not intersect, then Morse index of any saddle equilibrium equals $1$ or $(n-1)$, that is $f^t\in G(\mathcal{S}^n_g)$. Moreover, there exists $k\geq0$ such that $\nu_{f^t} = 2g+k$ and $\mu_{f^t}=k+2$.
For case $g=0$ Theorem 2 is proved in [4], where necessary and sufficient conditions of topological equivalence of flows from class $G(S^n)$, $n\geq 3$, where obtained. Theorem 2 allows to obtain topological classification of flows from class $\mathcal{S}^n_g$, $g>0$, in combinatorial terms using techniques of [4, 3].
For any $f^t\in G(\mathcal{S}^n_g)$ we put in correspondence a bicolor graph $\Gamma_{f^t}$ which describes mutual arrangement of invariant manifolds of saddle equilibria of the flow $f^t$, and provide the following result.
Theorem 3. Flows $f^t, {f'}^r\in G(\mathcal{S}^n_g)$ are topological equivalent iff their bicolor graphs $\Gamma_{f^t}, \Gamma_{{f'}^t}$ are isomorphic by means preserving colors isomorphism.
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