I'm reading through examples of computing Euler characteristic of manifolds. I know how to compute it for generic manifolds like sphere and torus. But what about matrix manifolds? I'd like to know how to compute the Euler characteristic of a matrix group, say $SL_3(\mathbb{R})$, for example.
What I know: The definition of Euler characteristic for a manifold $M$, I'm using is $\chi(M)=L(Id)$, where $L$ is the Lefschetz number of the identity map on $M$, which is basically the intersection number of the diagonal of the identity with itself. I also know the Poincare-Hopf theorem.
Any help is appreciated. Thanks!
Best Answer
You need to pass to something compact first so that we may apply the Lefschetz fixed point theorem.
1) The Euler characteristic is a homotopy invariant.
2) Every connected Lie group has a compact subgroup that it deformation retracts onto. For $SL_n$ it is $SO(n)$: this is a continuous version of the Gram-Schmidt procedure.
3) Now, and only now, may we apply Lefshcetz: Pick any non-identity elemeny of your connected compact group $G$. Left multiplication $L_g$ by that element is a continuous map with no fixed points, so $L(L_g) = 0$. Picking a path from $g$ to the identity $e$ gives a homotopy between $L_g$ and $L_e = \text{Id}$. Because the Lefschetz number is (defined to be!) a homotopy invariant, $L(L_e) = \chi(G) = 0$.