The suspension isomorphism for cohomology index braids

Abstract

Let $X$ be a metric space, $\pi$ be a local semiflow on $X$, $k\in{\mathbb N}$, $E$ be a $k$-dimensional normed real vector space and $\widetilde\pi$ be the semiflow generated by the equation $\dot y=Ly$, where $L\colon E\to E$ is a linear map whose all eigenvalues have positive real parts. We show in this paper that for every admissible isolated $\pi$-invariant set $S$ there is a well-defined isomorphism of degree $k$ from the (Alexander-Spanier)-cohomology categorial Conley-Morse index of $(\pi,S)$ to the cohomology categorial Conley-Morse index of $(\pi\times\widetilde\pi,S\times\{0\})$ such that the family of these isomorphisms commutes with cohomology index sequences. This extends previous results by Carbinatto and Rybakowski to the Alexander-Spanier-cohomology case.