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.
