= Homotopy invariance of mapping degree
{title2=$f_0\simeq f_1\Longrightarrow\deg f_0=\deg f_1$}
For maps between oriented connected <closed manifolds> of the same dimension, a <homotopy> induces equal maps on top-dimensional <homology>, hence equal coefficients on the <fundamental class>. For smooth maps and a smooth homotopy, the <degree by integration of a pullback volume form> gives another proof: the <Generalized Stokes theorem> on the homotopy cylinder makes the difference of endpoint pullback integrals zero because the target top form is closed. This is an obstruction to extending a degree-one sphere identity over a ball.
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