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!
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, $L(L_e) = \chi(G) = 0$.