Index Filtrations and the Homology Index Braid for Partially Ordered Morse Decompositions

Abstract

On a Morse decomposition of an invariant set in a flow there are partial orderings defined by the flow. These are called admissible orderings of the Morse decomposition. The index filtrations for a total ordering of a Morse decomposition are generalized in this paper with the definition and proof of existence of index filtrations for admissible partial orderings of a Morse decomposition. It is shown that associated to an index filtration there is a collection of chain complexes and chain maps called the chain complex braid of the index filtration. The homology index braid of the corresponding admissible ordering of the Morse decomposition is obtained by passing to homology in the chain complex braid.