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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!

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  • $\begingroup$ math.stackexchange.com/questions/13260/… $\endgroup$ – Zircht Dec 8 '18 at 2:29
  • $\begingroup$ @Zircht I have no knowledge of Lie theory. Most of what's mentioned there makes no sense to me. I'd like to use the definition I mentioned. There has to be a simpler way. $\endgroup$ – tangentbundle Dec 8 '18 at 2:36
  • $\begingroup$ The tangent bundle of a Lie group is parallelizable via left invariant vector fields. Hence it is trivial and has Euler number zero. $\endgroup$ – Charlie Frohman Dec 8 '18 at 3:09
  • $\begingroup$ @Zircht The tools used there are waaaaay too hard for this problem. You need those only if you care about calculating the entire homology groups. $\endgroup$ – user98602 Dec 9 '18 at 1:06
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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, $L(L_e) = \chi(G) = 0$.

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