On Singular Homology in Differentiable Spaces

Abstract

In a recent paper [4], Eilenberg proved that in a manifold of class Uk the study of singular simplexes and the study of singular simplexes of class k in fact, we shall prove the theorem for a more general category of spaces, defined as follows. DEFINITION 1. A subset M of a euclidean space E' is said to be a differentiable space of class Ck if there exist an open set U of En containing M and a differentiable mapping 0: U -* M of class Ck such that 0(x) = x for each x e M. 0 is called a retraction of class Ck. It follows from a lemma of Whitney [5, p. 667] that every differentiable manifold M of class Ck in En is a differentiable space of class Ck. In the sequel, we shall assume M to be a given differentiable space of class Ck in the euclidean space En and 0 to be a given retraction of class Ck of some open set U onto M. LEMMA 1. Let X and Xo be closed bounded subsets of some euclidean space Em such that Xo C X. Let f: X -* M be a mapping which is of class Ck on Xo. Then there exists a homotopy ft: X M (O 0 such that the e-neighborhood N, of f(X) is contained in the open set U of En. According to Lemma 3 of Eilenberg [4, p. 673], there exists a mapping g: X -N. of class &k such that g(x) = f(x) for each x e Xo and that p(f(x), g(x)) < e for each x e X, where p denotes the distance function of En. Define a homotopy 4e: X -+ N. (O ? t < 1) by taking 4t(x) the point which divides the line-segment joining f(x) to g(x) in the ratio t:(1 t). Then the required homotopy ft: X -E M (O < t < 1) is given by ft = 0t . This completes the proof. Let X be a closed bounded subset of Em and 1 the closed interval (0, 1) of real numbers. Then the topological product X X I is a closed bounded subset of Em+l. DEFINITION 2. Two mappings f, g: X -* M of class Ck are said to be Ck -homotopic if there exists a mapping h: X X I -+ M of class Ck such that h(x, 0) = f(x) and h(x, 1) = g(x) for each x e X. LEMMA 2. Two mappings f, g: X -+ M of class Ck are Ck-homotopic if and only if they are homotopic. PROOF. The necessity is trivial. To prove the sufficiency, let us suppose