F-deformations and f-tractions

Abstract

Let M(n) be a compact, connected topological manifold and F a continuous mapping of M(n) into R that is "topologically nondegenerate" in the sense of (Morse, M. (1959) J. d'Analyse Math., 7, 189-208). Let c be a value of F and set F(c) = {pinM(n)|F(p) </= c}. The topological critical points of F on F(c) are finite in number and can be related to the invariants of the homology groups of F(c) as in the differentiable case. F-Deformations and F-tractions make this possible. F-Tractions are here defined and replace retracting deformations used in the differentiable case. The final relations will serve as a model for similar relations between critical extremals of Weierstrass integrals on Riemannian manifolds.