On the reduction of the topological classification of gradient-like flows problem to the classification of polar flows

I. A. Saraev

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 25 (2023) no. 2, pp. 62-75

Résumé

In this paper we consider a class $G(M^n)$ of gradient-like flows on connected closed manifolds of dimension $n\geq 4$ such that for any flow $f^t \in G(M^n)$ stable and unstable invariant manifolds of saddle equilibria do not intersect invariant manifolds of other saddle equilibria. It is known that the ambient manifold of any flow from the class $G(M^n)$ can be splitted into connected summ of the sphere $\mathbb{S}^n$, $g_{f^t} \geq 0$ copies of direct products $\mathbb{S}^{n-1} \times \mathbb{S}^1$, and a simply connected manifold which is not homeomorphic to the sphere. The number $g_{f^t}$ is determined only by the number of nodal equilibria and the number of saddle equilibria such that one of their invariant manifolds has the dimension $(n-1)$ (we call such equilibria trivial saddles). A simply connected manifold which is not homeomorphic to the sphere presents in the splitting if and only if the set of saddle equilibria contains points with unstable manifolds of dimension $i \in \{2,\dots,n-2\}$ (we call such equilibria non-trivial saddles). Moreover, the complete topological classification was obtained for flows from the class $G(M^n)$ without non-trivial saddles. In this paper we prove that for any flow $f^t\in G(M^n)$ the carrier manifold can be splitted into a connected sum along pairwise disjoint smoothly embedded spheres (separating spheres) that do not contain equilibrium states of the flow $f^t$ and transversally intersect its trajectories. The restriction of the flow $f^t$ to the complements to these spheres uniquely (up to topological equivalence and numbering) defines a finite set of flows $f^t_1, \dots, f^t_l$ defined on the components of a connected sum. Moreover, for any $j\in 1, \dots, l$, the set of saddle equilibria of the flow $f^t_j$ consists either only of trivial saddles or only of of non-trivial ones and then the flow $f^t_j$ is polar. We introduce the notion of consistent topological equivalence for flows $f^t_1,\dots f^t_j$ and show that flows $f^t, {f'}^t\in G(M^n)$ are topologically equivalent if and only if for each of these flows the set of separating spheres exists that defines consistently topologically equivalent flows on the components of the connected sum.

Détail

Keywords: gradient-like flows, manifold, topological classification, Morse-Smale flows, Morse function