The boundary map and the connecting orbits

Abstract

The purpose of this article is to show that the image of the homological boundary map attached to a filtration for an attractorrepeller pair of a smooth flow on a compact manifold is a submodule of the Alexander cohomology of certain order of the connecting set (some restrictions have to be imposed in order to have a valid argument). In particular, this gives an affirmative answer to a conjecture in Conley index theory which states that if the boundary map is not zero in two dimensions, the connecting set cannot be contractible. The Conley index. Blocks. The connecting setSuch an invariant set S is called isolated invariant if there is a compact neighborhood N of S with the property that the maximal invariant set in N is S. The neighborhood itself is called an isolating neighborhood and its existence guarantees that S is compact. Describing invariant sets based on an algebraic or a topological invariant is an old idea which goes back to the works of Poincaré and Morse. Charles Conley introduced a general approach to this problem based on the notion of index pairs. Definition 1.1. Let S be an isolated invariant set for a continuous flow Φ in a locally compact metric space M. Then an index pair is a pair of compact sets (N 1 , N 0 ) in M such that:(1) N 0 ⊂ N 1 and N 1 \ N 0 is a neighborhood of S and S is the maximal invariant set in cl(N 1 \ N 0 );(2) (positive invariance of N 0 ) If x ∈ N 0 and x · [0, t] ⊂ N 1 for some t 0, then x · [0, t] ∈ N 0 ;(3) (exit set property of N 0 ) If x ∈ N 1 and x · [0, ∞) N 1 , then there exists a t 0 with x · [0, t] ⊂ N 1 and x · t ∈ N 0 .