Geometry & Topology

About the homological discrete Conley index of isolated invariant acyclic continua

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

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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.

Article information

Geom. Topol., Volume 17, Number 5 (2013), 2977-3026.

Received: 6 February 2013
Revised: 24 June 2013
Accepted: 26 June 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37B30: Index theory, Morse-Conley indices 37C25: Fixed points, periodic points, fixed-point index theory 54H25: Fixed-point and coincidence theorems [See also 47H10, 55M20]

fixed point index Conley index filtration pairs


Hernández-Corbato, Luis; Le Calvez, Patrice; R Ruiz del Portal, Francisco. About the homological discrete Conley index of isolated invariant acyclic continua. Geom. Topol. 17 (2013), no. 5, 2977--3026. doi:10.2140/gt.2013.17.2977.

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