Restriction from continuous to smooth singular cochains

Let $M$ be a smooth manifold, possibly with boundary. The inclusion $I_M:C_{\bullet }^{\mathrm{\infty }}(M;\mathbb{R})\hookrightarrow C_{\bullet }(M;\mathbb{R})$ is a chain map by Smooth singular chain and cochain complexes. Define the restriction comparison by ${\rho }_M^k:C^k(M;\mathbb{R})\to C_{\mathrm{\infty }}^k(M;\mathbb{R}),{\rho }_M^k(\phi )=\phi \circ I_M\mathrm{.}$ On each smooth simplex this retains exactly the original cochain value. It is real-linear and satisfies $\delta {\rho }_M\phi =\phi I_M\mathrm{\partial }=\phi \mathrm{\partial }{I}_M={\rho }_M\delta \phi$. It therefore sends cocycles to cocycles and coboundaries to coboundaries, inducing $H^k({\rho }_M):H_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^k(M;\mathbb{R})\to H_{\mathrm{\infty }}^k(M;\mathbb{R})$ on the quotients in Real singular cohomology.

For a smooth map $f:M\to N$, both composites $f_{\mathrm{\#}}{I}_M$ and $I_N{f}_{\mathrm{\#}}^{\mathrm{\infty }}$ send a smooth simplex $\sigma$ to $f\sigma$. Hence restriction commutes with pullbacks, on cochains and on their quotient classes. No choice of a smooth approximation is involved. Negative groups and empty-manifold groups are zero; on a point all simplices are smooth and $\rho$ is the identity in every nonnegative cochain degree. This definition does not yet assert that $H({\rho }_M)$ is an isomorphism.