10 The Work of Stephen Smale in Differential Topology MORRIS W. HIRSCH Background The theme of this conference is “Unity and Diversity in Mathematics.” The diversity is evident in the many topics covered. Reviewing Smale's work in ...

Assume n,k,m,q are positive integers. Let M^n denote a smooth differentiable n-manifold and R^k Euclidean k-space. (a) If M^n is open it imbeds smoothly in R^k, k=2n-1 (b) If M^n is open and parallelizable it immerses in R^n (c) Assume M^n is closed and (m-1)-connected, 1< 2m-n < n+1. If a neighborhood of the (n-m)-skeleton immerses in R^q, a>2n-2m, then the complement of a point of M^n imbeds smoothly in R^q.

If K is a subcomplex of a smooth triangulation of a manifold M and N is a regular neighborhood of K contained in an open neighborhood U of K in M, there is a piecewise regular homeomorphism f of M carrying N onto a smooth submanifold of U, and f reduces to the identity map on K and outside U.

Let phi be a flow on a compact metric space A and let p be a chain recurrent point. We show that H'(A; R) ^ 0 if dim_ p (A) = 1 or p belongs to a section of phi. Applications to planar flows and to smooth flows are given.

I discuss old and new results on fixed points of local actions by Lie groups G on real and complex 2-manifolds, and zero sets of Lie algebras of vector fields. Results of E. Lima, J. Plante and C. Bonatti are reviewed.