The integral homology of Smale diffeomorphisms (Q1077775)

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.