In the GTM218,Theorem 18.7:
For any smooth manifold $M$,the map $l_*:H_p^{\infty}(M)\rightarrow H_p(M)$ induced by inclusion is an isomorphism.
Is it also true for smooth manifold with boundary?(It may be wrong because it uses the Whitney approximation theorem) I want to know this because I want to define the de Rham homomorphism on smooth manifold with boundary,but the book only deals with smooth manifold without boundary.(And Stokes' theorem for chains only deals with smooth manifold without boundary too.)
I think that's right, otherwise it's disappointing.