The suspension isomorphism for homology 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
space and $\widetilde\pi$ be the semiflow generated by the
equation $\dot y=Ly$, where $L\co E\to E$ 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 homology categorial Conley-Morse index
of
$(\pi\times\widetilde\pi,S\times\{0\})$ to the homology categorial
Conley-Morse index of $(\pi,S)$ such that the family of these
isomorphisms commutes with homology index sequences. In
particular, given a partially ordered Morse decomposition
$(M_i)_{i\in P}$ of $S$ there is an isomorphism of degree
$-k$ from the homology index braid of
$(M_i\times\{0\})_{i\in P}$ to the homology index braid of
$(M_i)_{i\in P}$, so $C$-connection matrices of
$(M_i\times\{0\})_{i\in P}$ are just $C$-connection
matrices of $(M_i)_{i\in P}$ shifted by $k$ to the
right.