The Space of Framed Functions

We define the notion of a "framed function" on a compact smooth manifold N and we show that the space of all framed functions on N is (dim N I)-connected. A framed function on N is essentially a smooth function N --> R with only Morse and birth-death singularities together with certain additional structure. Introduction. The purpose of this paper is to prove that the space of framed functions on a compact smooth manifold N is (dim N I)-connected. The precise definitions and statements are given in §1. The purpose of this introduction is to explain why we are interested in framed functions, in particular we explain how they are related to pseudoisotopy theory, Suppose that wn+l is a cobordism from Mo to Ml (all manifolds being compact and smooth). Thus oW = Mo U Ml U (oMo X 1). Also suppose we have a Morse function f: W -> [, Then we get a finite relative cell complex X = Mo U e1 U e2 U ". U em homotopy equivalent to W with one cell for each critical point of I, the dimension of the cell being equal to the index of the critical point. There is however some ambiguity in the assignment of X to the function f. In order to resolve this ambiguity we add extra structure to the function I and the result is what we call a "framed Morse function," The first bit of extra structure that we add is a Riemannian metric on W with the appropriate boundary conditions. Given such a metric the cell e(xo) corresponding to a critical point Xo of I can be defined to be the closure of the set of all yEW so that the trajectory Ht) of 'VI through y converges to Xo as t -> + 00 (although we usually use a different more complicated definition for the cells e( x». Given the cells of X as point sets we still need a parametrization of the cells, i.e. for each closed i-cell ei we need a map from the standard i-disk Di into X with image e i• To achieve this it is enough to choose an orthonormal basis for the tangent space of the cells e(x) at each of the critical points x of f. The tangent space Txe(x) is the same as the (-) eigenspace of D 2/(x) considered as the covariant derivative of 'VI at x. Received by the editors November 18, 1985. Presented in a talk at the NSF-CBMS Conference on Geometric Topology held at the University of Notre Dame, July 16-27,1984. 1980 Mathematics Subject Classification (1985 Revision). Primary 57R65; Secondary 57R45.