The Differentiable Chain Functor Is Not Homotopy Equivalentto The Continuous Chain Functor

摘要

Let S_* and S_*~∞ be the functors of continuous and differentiable singular chains on the category of differentiable manifolds. We prove that the natural transformation i:S_*~∞→S_* , which induces homology equivalences over each manifold, is not a natural homotopy equivalence.