Morse index of saddle equilibria of gradient-like flows on connected sums of ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$

Abstract

Let $M$ be either $n$-sphere ${\mathbb{S}}^n$ or a connected sum of finitely many copies of ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$, $n\ge 4$. A flow $f^t$ on $M$ is called gradient-like whenever its non-wandering set consists of finitely many hyperbolic equilibria and their invariant manifolds intersects transversally. We prove that if invariant manifolds of distinct saddles of a gradient-like flow $f^t$ on $M$ do not intersect each other (in other words, $f^t$ has no heteroclinic intersections), then for each saddle of $f^t$ its Morse index (i.e. dimension of the unstable manifold) is either $1$ or $n-1$, so there are no saddles with Morse indices $i\in \{2,\dots ,n-2\}$.

4 pages, no figures