About the homological discrete Conley index of isolated invariant acyclic continua

TL;DR

This paper provides a detailed analysis of the discrete Conley index for isolated acyclic invariant sets under local homeomorphisms, revealing its periodic nature and implications for fixed point indices in three-dimensional space.

Contribution

It offers a complete description of the first discrete homological Conley index for acyclic continua, establishing its periodicity and deriving new fixed point index bounds.

Findings

The first discrete homological Conley index is periodic.

Fixed points of orientation-reversing homeomorphisms in R^3 have index ≤ 1.

No minimal orientation-reversing homeomorphisms exist in R^3.

Abstract

This article includes an almost self-contained exposition on the discrete Conley index and its duality. We work with a local homeomorphism of $\mathds{R}^d$ and an invariant and isolated acyclic continuum, such as a cellular set or a fixed point. In this setting, we obtain a complete description of the first discrete homological Conley index, which is periodic, that enforces a combinatorial behavior of higher indices. As a consequence, we prove that isolated (as an invariant set) fixed points of orientation-reversing homeomorphisms of $\mathds{R}^3$ have fixed point index $\le 1$ and, as a corollary, that there are no minimal orientation-reversing homeomorphisms in $\mathds{R}^3$.