The suspension isomorphism for homology index braids - Carbinatto, Maria C.; Rybakowski, Krzysztof P. (Topological Methods in Nonlinear Analysis, 2006)
Let $X$ be a metric space, $π$ be a localsemiflow on $X$, $k\in\mathbb N$, $E$ be a $k$-dimensional normedspace and $\widetildeπ$ be the semiflow generated by theequation $\dot y=Ly$, where $L\co E→ E$ is a linear mapwhose all eigenvalues have positive real parts. We show inthis paper that for every admissible isolated$π$-invariant set $S$there is a well-defined isomorphism of degree $-k$ fromthe homology categorial Conley-Morse index of$(π×\widetildeπ,S×\{0\})$ to the homology categorialConley-Morse index of $(π,S)$ such that the family of theseisomorphisms commutes with homology index sequences. Inparticular, 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$(M_i×\{0\})_{i\in P}$ to the homology index braid of$(M_i)_{i\in P}$, so $C$-connection matrices of$(M_i×\{0\})_{i\in P}$ are just $C$-connectionmatrices of $(M_i)_{i\in P}$ shifted by $k$ to theright.