# Notions of degree for maps $S^n \to S^n$?

In algebraic topology, we define a degree for a map $f: S^n \to S^n$ as where the induced map $f_*$ on the $n$-th homology group of $S^n$ sends $1$.

In differential topology, we have a different (same?) notion of degree for $f$. You take a regular value $b \in S^n$, consider $f^{-1} (b)$ (which is finite by the inverse function theorem and some compactness argument), and take the difference between the number of points in the preimage where the Jacobian of $f$ is positive and the number of points in the preimage where the Jacobian of $f$ is negative.

Geometrically, I can see that they are the same, but I couldn't convince myself rigorously. In Prop 2.30 of Hatcher, he mentions that the degree of $f$ is the sum of the local degrees of $f$ at each preimage point, and local degrees are either $\pm 1$. (Local degree is defined in the middle of page 136 in Hatcher.)

So, the final question is, must the sign of the local degree of $x \in f^{-1}(b)$ the same as the sign of the Jacobian of $f$ at $x$?

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There's technical details -- the jacobian needs to be non-degenerate for the formula to hold. But yes, they're the same. This is covered in Milnor's "topology from a differentiable viewpoint", Guillemin and Pollack's "differential topology", and it's also in Bredon's "geometry and topology". This construction can be viewed as something of a first step in the Pontriagin construction, covered in more detail in the Milnor text. –  Ryan Budney Nov 11 '09 at 9:17
@Ryan: why didn't you make this an answer? –  Andrew Stacey Nov 11 '09 at 12:39
The comment feature somehow seems more humble. I guess I'm getting unnaturally attached to it. –  Ryan Budney Nov 11 '09 at 19:13

I think what you need is the following lemma (usually called the "Stack of records" lemma):

Consider a smooth proper map of manifolds of the same dimension $f \colon M \to N$ and let $y \in N$ be a regular value of $f$.

Then there exists a neighbourhood $V \subset N$ of $y$ such that $f^{-1}(V) = \cup\_{i=1}^n U\_i$ with $U\_i \cap U\_j = \emptyset$ for $i \neq j$ and $f|\_{U_i} \colon U\_i \to V$ is a diffeomorphism for all $i$.

Now from this you can just sum up $\pm 1$ according to orientation on each $U_i$ to get the local degree of $f$ at $y$, and this works for both definitions of degree.

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I think so: it looks like the local degree according to Hatcher's definition measures whether $f$ preserves orientation or reverses it on the neighborhood of $x$. On page 233 he begins discussion of orientation using excision classes: an orientation for an neighborhood in an $n$-manifold at a point $x$ is just a choice of generator of $H_n(\mathbb R^n, \mathbb R^n-x)$, and a small neighborhood $U$ about $x$ is homeomorphic to $\mathbb R^n$. In his degree-counting, he takes a neighborhood $U$ of $x$ which is disjoint from other preimages $f^{-1}(f(U))$ and looks at the sign of the map $H_n(U, U-\{x\})\rightarrow H_n(f(U), f(U)-\{y\})$.
The sign of the Jacobian of $f$ should also tell you whether $f$ is locally orientation-preserving or reversing at at $x$.