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
Publication
Received: 6 February 2013
Revised: 24 June 2013
Accepted: 26 June 2013
Published: 17 October 2013
Proposed: Yasha Eliashberg
Seconded: Leonid Polterovich, Steve Ferry
Authors
 Luis Hernández-Corbato Departamento de Geometría y Topología Universidad Complutense de Madrid Plaza de Ciencias 3 28040 Madrid Spain Patrice Le Calvez Institut de Mathematiques de Jussieu Université Pierre et Marie Curie UMR 7586 CNRS Case 247, 4, place Jussieu 75252 Paris France Francisco R Ruiz del Portal Departamento de Geometría y Topología Universidad Complutense de Madrid Plaza de Ciencias 3 28040 Madrid Spain http://www.mat.ucm.es/~rrportal/