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.

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

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