Does de Rham theorem hold for manifolds with boundary?

differential-topology homology-cohomology smooth-manifolds

Yes, it holds for manifolds with boundary. One way to see this is to note that if $M$ is a smooth manifold with boundary, then the inclusion map $\iota\colon \text{Int}\ M\hookrightarrow M$ is a smooth homotopy equivalence (Thm. 9.26 in ISM), which therefore induces isomorphisms on de Rham cohomology (Thm. 17.11). Since $\iota$ also induces isomorphisms on singular cohomology, and the de Rham homomorphism (integration over chains) commutes with inclusion, the result follows.

(It would have been a good idea to include this in the book, either as a corollary or as a problem. I don't know why I didn't.)

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