A shape theory with singular homology
Friedrich-Wilhelm Bauer · Pacific Journal of Mathematics · 1976
A modified concept of a shape of a topological space is introduced which allows some basic geometric constructions: (1) One has a convenient homotopy concept which originates from a cylinder functor.(2) All inclusions of compact metric spaces are cofibrations.(3) Shape mappings which agree on the intersection of their counterimages can be pasted together (existence of push-outs).(4) There exists a singular complex S which has the same properties for shape mappings as the ordinary singular complex S for continuous maps.(5) Consequently one has a singular (shape) homology which for compact metric spaces turns out to be isomorphic to the (shape-theoretically defined) homotopical homology (in the sense of G. W. Whitehead) and to the Steenrod-Sitnikov homology.