Degree of Gauss map equal to half the Euler characteristic and Poincaré-Hopf

Question

The Poincaré-Hopf theorem states that for a smooth compact $m$-manifold $M$ without boundary and a vector field $X\in\operatorname{Vect}(M)$ of $M$ with only isolated zeroes we have the equality $$\sum_{\substack{p\in M\\X(p)=0}}\iota(p,X)=\chi(M)$$ where $\iota(p,X)$ denotes the index of $X$ at $p$ and $\chi(M)$ denotes the Euler characteristic of $M$.

Let $m$ be even and $M\subset\mathbb{R}^{m+1}$ be a $m$-dimensional smooth compact submanifold without boundary and denote by $\nu:M\to S^m$ its Gauss map. How can I deduce from the Poincaré-Hopf theorem that the Brouwer degree of $\nu$ is equal to half the Euler characteristic of $M$ i.e. $$\deg(\nu)=\frac{1}{2}\chi(M)?$$

After many repeated (unsuccessful) tries I was hoping hat someone else might shed some light onto this...

Answer 1

One way to go about this is to start with your manifold $M \subset \mathbb R^{m+1}$ and consider a height function $f : \mathbb R^{m+1} \to \mathbb R$ (orthogonal projection onto a vector) restricted to $M$. So there is some fixed vector $v \in S^m$ such that $f(x)=\langle x,v\rangle$ for al $x \in \mathbb R^{m+1}$.

Generically, this is a Morse function so its gradient (the orthogonal projection of $v$ to $TM$) is a vector field which is transverse to the $0$-section of $TM$. So Poincare-Hopf tells you how you can compute the Euler characteristic from this.

Now how is that related to the Gauss map? If you were to compute the degree of the Gauss map $\nu : M \to S^m$ by computing its intersection number with $v \in S^m$ you would have a very similar looking sum to compute! But do notice that the orthogonal projection of $v$ to $TM$ can be zero at both $\nu^{-1}(v)$ and $\nu^{-1}(-v)$. When you work out the details this ultimately explains why there's the $1/2$ and why it only works in even dimensions.

I hope that gives you the idea without giving too much away.

Answer 2

I like Ryan Budney's proof very much. For posterity, I will add my own, different proof. I will use the Jordan-Brouwer separation theorem (see p. 89 of the book by Guillemin & Pollack), the Poincaré-Hopf theorem for manifolds with boundary (which assumes that the vector field is outward pointing at the boundary), and the fact that the Euler characteristic of any odd-dimensional smooth closed manifold vanishes. (In the orientable case, the latter claim follows from Poincaré duality and is exercise 18-9 in Lee's Intro. to Smooth Manifolds, 2nd ed.; the nonorientable case follows from the orientable case, consideration of the orientation double cover, and the fact that the Euler characteristic of a degree $n<\infty$ covering space $E$ of a compact manifold $B$ with boundary satisfies $\chi(E) = n\cdot \chi(B)$. Alternatively, see Cor. 3.37 of Algebraic Topology by Hatcher.)

Proof: Let $M\subset \Bbb{R}^{m+1}$ be a smooth closed hypersurface. By the Jordan-Brouwer separation theorem, $M=\partial N$ is the boundary of some compact, smooth, codimension-$0$ manifold $N$ with boundary $\partial N$. Let $X$ be a smooth vector field on $N$ which points strictly outward at $\partial N$ and has isolated zeros in $\text{int}(N)$. By the Poincaré-Hopf theorem, the Euler characteristic $\chi(N)$ is equal to the sum of the indices of these zeros. On the other hand, the degree of the Gauss map $M\to S^m$ is also equal to the sum of these indices. (To see this, let $N'$ be $N$ minus the union of small open balls centered at the zeros of $X$; the Gauss map $\frac{X}{|X|}:\partial N'\to S^m$ admits a smooth extension to all of $N'$ and therefore has degree zero, as shown in Guillemin & Pollack.) Hence

$\chi(N) = \text{degree of the Gauss map }M\to S^m.$ (1)

Now, in general the following Euler characteristic formula holds for smooth closed manifolds $N$ with nonempty boundary:

$\chi(DN) = 2\chi(N)-\chi(\partial N),$ (2)

where $DN$ is the (compact, boundaryless) double of $N$ obtained by pasting two copies of $N$ together along their boundaries and smoothing the result. When $\dim(N)=\dim(DN)$ is odd, $\chi(DN)=0$ as mentioned in the preface.

Therefore, in our case when $\dim(M) = \dim(\partial N)$ is even, the left-hand side of (2) vanishes to yield, when combined with (1),

$\frac{1}{2}\chi(M)= \text{degree of the Gauss map }M\to S^m$

when $\dim(M)$ is even.

This proves the desired result.

Bonus: From (1) and (2) we also obtain the following general formula, valid for arbitrary $\dim(M) = \dim(\partial N)$:

$\chi(N)=\frac{1}{2}[\chi(M)+\chi(DN)]= \text{degree of the Gauss map }M\to S^m$

where again $M = \partial N$ and the existence of $N$ is guaranteed by the Jordan-Brouwer separation theorem. Hence if $\dim(M)=\dim(\partial N)$ is odd, then $\chi(M)=0$ so that

$\chi(N)=\frac{1}{2}\chi(DN)= \text{degree of the Gauss map }M\to S^m$

when $\dim(M)$ is odd.