Integral cohomology classes define real de Rham periods

Assume countable choice. For every finite-dimensional Hausdorff second-countable smooth manifold $Y$ without boundary, coefficient inclusion $\mathbb{Z}\hookrightarrow \mathbb{R}$ and the standard comparisons define a group homomorphism ${\rho }_Y:H_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^4(Y;\mathbb{Z})⟶H_{\mathrm{d}\mathrm{R}}^4(Y;\mathbb{R})\mathrm{.}$ Concretely, extend an integral singular cocycle coefficientwise to a real cocycle, pass from continuous to smooth singular cohomology by restriction, and apply the inverse of the de Rham integration isomorphism. This composite is well defined and sends every torsion class to zero.

Hatcher, Algebraic Topology, §3.1

The displayed map records only the real cohomology class of an integral class. In particular, it discards torsion and does not provide differential-cohomology data.

Here $H_{\mathrm{\infty }}^4(Y;\mathbb{R})$ denotes smooth singular cohomology as in [F3]–[F5]. The rows point to the right; the upper arrows are coefficient extension $r_Y$ and restriction $s_Y$, the vertical arrow is ${\rho }_Y$, and the lower arrow is de Rham integration $I_Y$. The diagram records $I_Y\circ {\rho }_Y=s_Y\circ r_Y$; the proof above establishes this identity from the maps and isomorphism suppliers, rather than relying on the picture.