Coherent singular complexes in strong shape theory
1) The shape singular complex \(\bar S(\)X) is constructed in the same way as the ordinary singular complex S(X) simply by replacing continuous mappings \(\sigma^ n: \Delta^ n\to X\) by strong shape mappings. It has virtually the same universality properties as S(X) (using strong shape mappings instead of continuous mappings). Moreover \(\bar S\) turns out to be a functor from the strong shape category (being defined appropriately) into the category of Kan complexes. 2) Once this is understood, one is enabled to introduce strong shape singular homology \(H_*(\bar S(X))=\tilde H_*(X)\). Strong shape homotopy groups \({\bar \pi}{}_ n(X,x_ 0)\) are isomorphic to \({\bar \pi}{}_ n(\bar S(X),*)\). Many well-known results (like a Hurewicz- theorem) are in the strong shape category immediate consequences of their classical counterparts (for the category Top). It turns out that \(\tilde H_*(X)\) is (because of the absence of a cluster axiom) generally not isomorphic to the Steenrod-Sitnikov homology groups (which, by the way, are isomorphic to some other kind of strong shape homology). All this is only a part of a, in the meantime, rather well-developed and in the literature broadly documented algebraic topology in strong shape theory (including applications in functional analysis, K-theory and bordism theory as well as an axiomatic classification of homology theories in strong shape theory for compact metric spaces). In order to be able to accomplish all that, one needs an adequate definition of a strong shape category (for compact metric spaces all approaches to strong shape theory have been proven to be equivalent on the homotopy level). Basically the present author rediscovers 1) and 2) in a setting somewhat different from that in the literature.
scientific article; zbMATH DE number 3899742