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.