#### Volume 17, issue 5 (2013)

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About the homological discrete Conley index of isolated invariant acyclic continua

### Luis Hernández-Corbato, Patrice Le Calvez and Francisco R Ruiz del Portal

Geometry & Topology 17 (2013) 2977–3026
##### Abstract

This article includes an almost self-contained exposition on the discrete Conley index and its duality. We work with a locally defined homeomorphism $f$ in ${ℝ}^{d}$ and an acyclic continuum $X$, such as a cellular set or a fixed point, invariant under $f$ and isolated. We prove that the trace of the first discrete homological Conley index of $f$ and $X$ is greater than or equal to $-1$ and describe its periodical behavior. If equality holds then the traces of the higher homological indices are 0. In the case of orientation-reversing homeomorphisms of ${ℝ}^{3}$, we obtain a characterization of the fixed point index sequence ${\left\{i\left({f}^{n},p\right)\right\}}_{n\ge 1}$ for a fixed point $p$ which is isolated as an invariant set. In particular, we obtain that $i\left(f,p\right)\le 1$. As a corollary, we prove that there are no minimal orientation-reversing homeomorphisms in ${ℝ}^{3}$.

##### Keywords
fixed point index, Conley index, filtration pairs
##### Mathematical Subject Classification 2010
Primary: 37B30, 37C25, 54H25