Vol. 54, No. 1, 1974

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On the homotopy invariance of certain functors

Brian Kirkwood Schmidt

Vol. 54 (1974), No. 1, 245–256

A functor from the category of topological spaces to the category of groups is said to be homotopy invariant if it carries homotopic mappings to the same mapping. It is well known, for example, that the homology and homotopy functors are homotopy invariant. On the other hand, the functor which takes each topological space M to the free abelian group generated by the points of M is not homotopy invariant. It will be shown that a functor which is not homotopy invariant must take topological spaces to groups which are very “large”. For example, the homology groups of a simplicial complex are finitely generated, while the free abelian group generated by the poin ts of a typical simplicial complex is uncountably generated. Among other results, it will be shown that every functor from simplicial complexes to finitely generated groups is homotopy invariant.

Mathematical Subject Classification
Primary: 55E05
Received: 21 February 1973
Published: 1 September 1974
Brian Kirkwood Schmidt