On Homotopy Type and Deformation Retracts

Abstract

It has recently been shown by J. H. C. Whitehead' that two complexes X and Y belong to the same homotopy type2 if and only if there is a third complex W of which both X and Y are deformation retracts.3 I shall show that this theorem holds not merely for complexes but for the most general spaces for which continuity has a meaning. The proof which I give is direct and constructive and avoids the extraneous notions of relative homology and relative homotopy groups which complicate Whitehead's proof. The concept of homotopy type splits naturally into two concepts which I shall call rightand left-homotopy inversion. In theorems 3.3 and 3.4 I show that right-and-left inversion correspond respectively to deformation and retraction, thus replacing Whitehead's theorem by two component theorems. The necessary preliminary study of deformation, retraction, and inversion is carried out in ??1 and 2, and the mapping cylinder, the fundamental tool of our theory, is defined in ?3. It should be noted that Whitehead's definition4 of mapping cylinder is not really satisfactory for the general spaces considered here. The fundamental theorems of this paper are theorems 3.1 and 3.2. They are generalizations of the theorems (3.3 and 3.4) discussed above. In ?4 these fundamental theorems are applied (in another direction) to the Hopf-Pannwitz deformation and also to yield a new characterization of the closure of a homogenous n-dimensional polyhedron. These theorems (3.1 and 3.2) are of considerable interest in themselves. They exhibit a duality which is quite striking and seem to indicate a relatively unexplored region which I might designate as algebra of mapping classes. In this connection they should be compared with the fundamental theorem of fibre spaces5 to which they bear an evident analogy. In ??5 and 6 certain specializations are considered. They are to be regarded as trends in the following two directions (a) bridging the gap between homotopy type and nucleus' (b) bridging the gap between homotopy type and topological type. In ?7 I develop an n-dimensional analogue of ?3. This is in line with