Topological non-degenerate functions
John C. Cantwell · Tohoku Mathematical Journal · 1968
Introduction.The theory of C°° non-degenerate functions has been useful in the study of differentiable manifolds.The theory of topological non-degenerate functions is much less developed.Morse [5] has established the Morse inequalities.Kuiper [3] has shown that any compact w-manifold which admits a topological non-degenerate function with two critical points is homeomorphic to S n .Eel Is and Kuiper [2] have studied compact manifolds which admit a topological non-degeneiate function with three critical points.In this paper we prove: THEOREM 1.1.Suppose f is a topological non-degenerate function defined on a compact n-manίfold.If [a, b] is an interval of regular values of f then f~l(a)x(Q,V) is homeomorpίτic to f~l(b)x(0 y ΐ).THEOREM 1.2.Suppose a compact n-manifold M admits a topological non-degenerate function f such that all the critical points of f of index λ lie at the level λ.Then M admits a cell decomposition with exactly as many cells of dimension λ as f has critical points of index λ.Theorem 1.1 illustrates some of the difficulties in the theory of topological non-degenerate functions.If [α, b] is an interval of regular values of a C°°n on-degenerate function, it is easy to show that f~l(a) is homeomorphic to f~l(b) (Milnor [4], p. 12).Whether or not this is true in the topological case is an open question.Theorem 1.2 is a partial solution of a problem of Eel Is and Kuiper [2], p. 195.We intend to give a more complete solution to this problem in a future paper. Notation and Terminology.We refer to Morse [5] as a general reference for §2 and §3.We denote cartesian w-space by R n .Let Nί = {(z ί9 -, z n }zR n \(z\+ ••• +$)*0.An n-manifold is a separable metric space each point of which has a neighbour-