Mapping Cylinders of Approaching Maps and Strong Shape

Abstract

is a diagram in HCM such that / is the homotopy class of a map /: Y-*• Z which* is both an embedding and a strong shape equivalence (these maps coincide with the SSDR-maps introduced in [3, 4]). Among all such decompositions of strong shape morphisms, 'mapping cylinder decompositions' play a distinguished role. These arise quite naturally when the morphisms of SSh are visualized as approaching homotopy classes of approaching maps [3,4,9]. The basic idea is to extend the mapping cylinder construction from ordinary maps to approaching maps. This was first done by S. Ferry in [7], utilizing resources from Hilbert cube manifold theory. In this paper we present a completely elementary alternative mapping cylinder construction (see Section 2 and the Basic Construction in Section 1). The essence of this construction is that it establishes a 1-1 correspondence between strong shape morphisms X-> Y and 'homotopy classes' of certain topologies on the disjoint union Xx. (0,1] + Y of the sets Xx (0,1] and Y; see Sections 3, 6. But now it can be shown that there exists a natural law of composition for such homotopy classes of topologies so that we obtain a category HO (the homotopy topology category whose objects are compacta and whose morphisms are the above homotopy classes of topologies) plus a category isomorphism yussh: SSh -• HO; see Section 5 and Theorem (6.2). This is a new and perhaps unexpected description of SSh which, moreover, supplies the above-mentioned mapping cylinder decompositions of strong shape morphisms (cf. Corollary (6.4)). Motivated by the mapping cylinder construction for approaching maps, we define an embedding/: Y -*• C of compacta to be a cylinder base embedding if there exists a compactum X such that C\J{Y) is homeomorphic to I x ( 0 , 1 ] . This is an internal geometric property of the pair (C,J{Y)) (we have already used this property in [11]). Our final result is that S: HCM -+ SSh localizes HCM at the collection ZCBE of