Exercise 7, § 2

Let X and Y be two real manifolds of class $C^r(1\leq r\leq \omega )$, separated, connected and of dimension $n$; let $f: X\rightarrow Y$ be a proper morphism of class $C^r$, equipped with an orientation F (Differentiable and Analytic Manifolds, Results, 10.2.5).

a) Show that there exists a unique real number $d$ such that $\int_Xf^*\alpha =d\int_Y\alpha$ for every twisted differential form $\alpha$ of degree $n$ on Y of class $C^1$ and of compact support (use Differentiable and Analytic Manifolds, Results, 11.2.4). b) Let $y\in Y$ be such that $f$ is étale at every point of $f^{-1}(y)$. For $x\in f^{-1}(y)$, put $\nu_x(f) = 1$ (resp. $\nu_x(f) =-1$) if the maps $F_x$ and $\widetilde{f}_x($Differentiable and Analytic Manifolds, Results, 10.2.5, Example $b))$ from Or(T$_x(X)$) to Or(T$_y(Y)$) coincide (resp. are opposite). Prove that $d=\sum_{x\in f^{-1}(y)}\nu_x(f)$ and, in particular, that $d\in \mathbf{Z}$.

We call $d$ the degree of $f$ and denote it by deg $f$. If X = Y and X is orientable, we can take for F the orientation that preserves the orientation of X.

c) Show that, if deg $f\not= 0$ then $f$ is surjective.

d) If there exists $x\in X$ such that $f^{-1}(f(x)) =\{x\}$ and such that $f$ is étale at $x$, then $f$ is surjective.