## Topological Methods in Nonlinear Analysis

### 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 $L\colon E\to E$ 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 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 28, Number 2 (2006), 199-233.

Dates
First available in Project Euclid: 13 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1463144816

Mathematical Reviews number (MathSciNet)
MR2289686

Zentralblatt MATH identifier
1127.37018

#### Citation

Carbinatto, Maria C.; Rybakowski, Krzysztof P. The suspension isomorphism for homology index braids. Topol. Methods Nonlinear Anal. 28 (2006), no. 2, 199--233. https://projecteuclid.org/euclid.tmna/1463144816

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