# Local degree computation (Hatcher exercise 2.2.8)

Let $f \in \mathbf{C}[z]$ be a polynomial considered as a map $\mathbf{C} \to \mathbf{C}$ and consider induced $\hat{f} : S^2 \to S^2$ on one-point-completion. Trying to compute local degree of $\hat{f}$ at a root $x$ ($f(x) = y$ is south pole of sphere), I finally reduced to following: let $U$ and $V$ be "small" neighbourhoods around $x$, $y$, resp, such that $f(U) \subset V$, and consider a map

$$(f|U)_* : H_2(U, U-\{x\}) \longrightarrow H_2(V, V-\{y\})$$

Those groups are isomorphic to $\mathbf{Z}$ by the excision etc. So this homomorphism is multiplication by some integer, which is local degree of $f$ at $x$ by definition. Now I want to show that this integer is the multiplicity $m$ of the root $x$. In fact if $U$ and $V$ are picked small enough, the restriction $f|U : U \to V$ is a $m$-sheeted cover, and intuitively I guess this should show the claim. But, I don't see the right argument. Actually I don't know what is a generator of either of these groups explicitly.

-