# $\Phi:M\to M$ be a smooth map whose restriction to $\partial M$ is the identity

Let $M$ be a connected oriented smooth n-dimensional manifold-with-boundary, and let $\Phi:M\to M$ be a smooth map whose restriction to $\partial M$ is the identity. If $\omega$ is any smooth n-form on $M$, prove that $\deg \Phi=1$, i.e. $$\int_M \Phi^* \omega = \int _M \omega$$

My attempt is to consider the homological sequence of the pair $(M, \partial M)$, and try to show $\Phi^*: H^n(M)\to H^n(M)$ is an isomorphism.

• This is an immediate consequence of Stokes' theorem. – user98602 Aug 22 '16 at 1:15
• Excuse me. Would you mind providing some hints? – Hang Aug 22 '16 at 1:23
• @MikeMiller Are you referring to the "classical" Stokes' Theorem or the general version? – cpiegore Aug 22 '16 at 1:42
• @cpiegore: The general version, of course. – Ted Shifrin Aug 22 '16 at 5:40
• Stoke's theorem can only apply to $d\omega$ over $M$ for some n-1 form, how does this work? – Hang Aug 22 '16 at 7:58