About the homological discrete Conley index of isolated invariant acyclic continua

Hernández-Corbato, Luis¹ ; Le Calvez, Patrice² ; R Ruiz del Portal, Francisco¹

¹ Departamento de Geometría y Topología, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain
² Institut de Mathematiques de Jussieu, Université Pierre et Marie Curie, UMR 7586 CNRS, Case 247, 4, place Jussieu, 75252 Paris, France

Geometry & topology, Tome 17 (2013) no. 5, pp. 2977-3026.

Résumé

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 ${ℝ}^3$.

Détail

DOI : 10.2140/gt.2013.17.2977

Classification : 37B30, 37C25, 54H25
Keywords: fixed point index, Conley index, filtration pairs

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