The differentiable chain functor is not homotopy equivalent to the continuous chain functor (Q1004054)

Let \(S_*\) and \(S_*^{\infty}\) be the functors of continuous and differentiable singular chains on the category of differentiable manifolds. For any manifold \(M\), the natural inclusion \(i_M: S_*^{\infty}(M) \to S_*(M)\) is defined. It was proved in \textit{S. Eilenberg} [Ann. Math. 48, No.~2, 670--681 (1947; Zbl 0029.41904)] that for any \(M\), there exists a chain map \(\Theta_M: S_*(M)\to S_*(M)^{\infty}\) which is a homotopy inverse to \(i_M\). In the paper under review, the authors prove that the natural transformation \(i:S_*^{\infty}\to S_*\), which induces homology equivalences over each manifold is not a natural homotopy equivalence. In fact, it is proved that there is no natural transformation of the functors \(S_*\to S_*^{\infty}\) that would induce isomorphisms in homology. Let \(S_*^{k}(M)\) be a complex of singular simplexes of class \(\mathcal C^k, k=0,1,2,\dots \). The authors prove also that there is no natural transformation of the functors \(S_*^{k}\to S_*^l\), with \(l>k\), that would induce isomorphisms in homology.