Vol. 64, No. 1, 1976

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A shape theory with singular homology

Friedrich-Wilhelm Bauer

Vol. 64 (1976), No. 1, 25–65

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.

Mathematical Subject Classification
Primary: 55D99, 55D99
Received: 3 May 1974
Published: 1 May 1976
Friedrich-Wilhelm Bauer