### 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.

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.

#### Keywords

Conley index; homology index braid; suspension isomorphism; connection matrix

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