Mapping degree of a real analytic map

Question

Let $X, Y$ be two compact, connected, orientable manifolds of equal dimension. For example $X = Y = \mathbb{CP}^n$. Suppose that $f \colon X \to Y$ is a surjective analytic map. Can we define the mapping degree of $f$, i.e. Does there exist an integer $d$ such that $\forall y \in Y$, $f^{-1}(y)$ has at most $d$ points?

Answer 1

Yes, you can define the degree as usual using differential topology (counting the preimage of a regular value) or algebraic topology (looking at the induced map on top homology or cohomology). However, no, for what you're asking, there need not exist such a $d$. Just take $Y$ to be any compact complex manifold and $X=\tilde Y$ the blow-up of $Y$ at a point $y_0$. Then the preimage of $y_0$ is a $\Bbb CP^{n-1}$. (You can do the same thing with a real blow-up of a smooth (or real analytic) manifold.)