Morse relations for curvature and tightness

Abstract

In chapter I we define the oritical points of a continuous function ϕ: X → IR on a compact metric space X, and the Morse polynomial for an isolated critical point. We reproduce the proof of the Morse relations for a function with isolated critical points for this more general set-up. In chapter II we define polynomial, absolute and alternating curvature measures of certain maps or embeddings of X into E^N in terms of critical points and we obtain Morse relations for these curvature measures. In chapter III we discuss maps (and sets) in E^N of minimal total absolute curvature (tight maps) and we recall some results on tight embeddings of smooth, piecewise linear and topological manifolds.

This is a considerably extended version of one of a series of four lectures given at the Liverpool Summer School on Singularities, June 1970.