The integral homology of Smale diffeomorphisms
type: Item
instance of: scholarly article
title: The integral homology of Smale diffeomorphisms
zbMATH Open document ID: 0595.58025
DOI: 10.1016/0040-9383(85)90052-7
published in: Topology
publication date: 1985-01-01T00:00:00Z
review text: The authors describe a way to deduce topological information for certain diffeomorphisms from their local dynamical information. The main results are given in two theorems. We first need some definitions. Let M be a closed connected orientable manifold and f a diffeomorphism of M satisfying axiom A. It is known that for such a diffeomorphism, the nonwandering set \(\Omega\) (f) can be decomposed into a finite disjoint union of basic sets \(\Omega_ 1,...,\Omega_ s\). Based on the asymptotic behavior of the points of the manifold under f, an ordering can be defined among these basic sets. The diffeomorphism f is said to have no cycles if the ordering can be extended into a total ordering of the \(\Omega_ i's\). It is also known that if the set \(\Omega_ i\) is zero dimensional it may have an integral matrix \(B_ i\) as its ''signed representative''. Now, let \(C_*\) be a free finitely generated Z-complex such that \(H_*(C_*)\cong H_*(M;Z)\), and let \(E: C_*\to C_*\) be a chain map. The pair \((C_*,E)\) is called a Z-endomorphism of f if there exists an isomorphism: \(H_*(C_*)\to H_*(M;Z)\) which conjugates \(E_*\) and \(f_*\). Finally, a fitted diffeomorphism is a Smale diffeomorphism which preserves a handle decomposition of the manifold. The two main theorems are: Theorem 1. Suppose f is a diffeomorphism of M satisfying axiom A, having no cycles, a zero dimensional \(\Omega\), and an indexing compatible ordering on the basic sets. For \(0\leq k\leq \dim (M)\), let \(B^ k_ 1,...,B^ k_{s_ k}\) be the signed representatives of the basic sets of index k, then there exists a Z-endomorphism \((C_*,E)\) of f such that for each k, \(E_ k\) is a nilpotent extension of \(B^ k_ 1,...,B^ k_{s_ k}.\) Theorem 2. Let M be a 2-connected manifold with torsion free homology, and dim \(M\geq 6\). If f is a diffeomorphism of Theorem 1 which has at least one fixed source and one fixed sink, then f is isotopic to an omega conjugate fitted diffeomorphism.
Mathematics Subject Classification ID: 37D15
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37C80
zbMATH DE Number: 3958288
zbMATH Keywords: chain complex
zbMATH Keywords: homology groups
zbMATH Keywords: Z-endomorphism
zbMATH Keywords: axiom A
zbMATH Keywords: nonwandering set
author: Michael Maller
author: Michael Shub
MaRDI profile type: Publication
full work available at URL: 0040-9383(85)90052-7
OpenAlex ID: W2017615568
description: scientific article; zbMATH DE number 3958288