Fitted diffeomorphisms of non-simply connected manifolds

Abstract

TOPOLOGICAL ENTROPY roughly speaking measures the complexity of a dynamical system. In “Homology Theory and Dynamical Systems”[ l] Shub and Sullivan defined a class of structurally stable diffeomorphisms called fitted which are Co dense in oil?‘(M). They show how to relate the dynamics of a fitted diffeomorphism to the induced chain map in homology theory, and using this determine the minimum topological entropy of fitted diffeomorphisms in each component of D%‘(M), for simply connected manifolds of dimension greater than five. In this paper we extend these results to the non simply-connected case. The main tools are a sharpened statement of the Whitney canceIling lemma for handles of index 2 and (n 2) using relative homotopy groups, and classical results of Whitelead concerning these groups[2]. Our methods also lead to an algebraic characterization of the chain complexes over Z[II,] which arise from handle decompositions of high dimensional manifolds, extending a theorem of Smale’s in the simply connected case. It is a pleasure to acknowledge many helpful conversations with Michael Shub and Dennis Sullivan.