Vol. 57, No. 2, 1975

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Homotopy invariance of contravariant functors acting on smooth manifolds

Brian Kirkwood Schmidt

Vol. 57 (1975), No. 2, 559–562

It is sometimes possible to prove that a functor is homotopy invariant using only a knowledge of the domain and range categories of the functor. It is known, for example, that every covariant or contravariant functor from the category of simplicial complexes (with continuous mappings) to the category of countable groups is homotopy invariant. This result has been extended to covariant, but not contravariant, functors with domain the category of smooth manifolds. In the contravariant case, the proof breaks down because certain mappings are not differentiable. This fault will be corrected in this paper. Among other results, it will be shown that every contravariant functor from the category of smooth manifolds to the category of countable groups is homotopy invariant.

Mathematical Subject Classification
Primary: 57A65
Secondary: 55E05
Received: 21 February 1973
Published: 1 April 1975
Brian Kirkwood Schmidt