### 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 $L\co E\to E$ 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 (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.

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 $L\co E\to E$ 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 (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.

#### Keywords

Conley index; Alexander-Spanier cohomology; cohomology index braid; suspension isomorphism

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