## Geometry & Topology

### About the homological discrete Conley index of isolated invariant acyclic continua

#### 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 $ℝd$ 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 $ℝ3$, we obtain a characterization of the fixed point index sequence ${i(fn,p)}n≥1$ 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 $ℝ3$.

#### Article information

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

Dates
Revised: 24 June 2013
Accepted: 26 June 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732693

Digital Object Identifier
doi:10.2140/gt.2013.17.2977

Mathematical Reviews number (MathSciNet)
MR3190304

Zentralblatt MATH identifier
1291.37021

#### Citation

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. https://projecteuclid.org/euclid.gt/1513732693

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