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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

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 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 3, we obtain a characterization of the fixed point index sequence {i(fn,p)}n1 for a fixed point p which is isolated as an invariant set. In particular, we obtain that i(f,p) 1. As a corollary, we prove that there are no minimal orientation-reversing homeomorphisms in 3.

fixed point index, Conley index, filtration pairs
Mathematical Subject Classification 2010
Primary: 37B30, 37C25, 54H25
Received: 6 February 2013
Revised: 24 June 2013
Accepted: 26 June 2013
Published: 17 October 2013
Proposed: Yasha Eliashberg
Seconded: Leonid Polterovich, Steve Ferry
Luis Hernández-Corbato
Departamento de Geometría y Topología
Universidad Complutense de Madrid
Plaza de Ciencias 3
28040 Madrid
Patrice Le Calvez
Institut de Mathematiques de Jussieu
Université Pierre et Marie Curie
Case 247, 4, place Jussieu
75252 Paris
Francisco R Ruiz del Portal
Departamento de Geometría y Topología
Universidad Complutense de Madrid
Plaza de Ciencias 3
28040 Madrid