I'm interested in methods of computing the homology groups and Euler characteristics of the classical Lie groups ($GL(n,\mathbb{R}), SL(n,\mathbb{R})$, etc.). (But I'd be interested in techniques which are more generally applicable to arbitrary Lie groups as well.)
Here's the current state of my knowledge on this issue with some explicit questions interspersed:
A compact Lie group of positive dimension admits a nowhere vanishing vector field and hence has Euler characteristic zero.
I remember reading in the past that a connected Lie group deformation retracts onto its maximal compact subgroup, which by the foregoing has Euler characteristic zero. Given this, we can say that a connected Lie group is either contractible or has Euler characteristic zero. However, my knowledge of Lie theory is (very) weak at best, so I am not fully comfortable accepting this fact. Is there a simple proof? (I feel that there should be but I'm just missing it.)
The Euler characteristic respects fiber bundle structures: if $F \rightarrow E \rightarrow B$ is a fiber bundle with total space $E$, base $B$, and fiber $F$, then $\chi(E) = \chi(F)\cdot\chi(B)$. Some suitable conditions are needed; I think $B$ path-connected suffices, but I'm not 100% certain. I can think of one explicit example of this: Identify $S^3$ with $SU(2)$ and implement the Hopf fibration $S^1 \rightarrow S^3 \rightarrow S^2$. The subgroup $U(1)$ is realized as $S^1$ and the quotient $SU(2)/U(1)$ is realized as $S^2$. So we have a fibration $SU(2) \rightarrow SU(2)/U(1)$ with fiber $U(1)$; hence $\chi(SU(2)) = \chi(SU(2)/U(1)) \cdot \chi(U(1)) = 0$. But this is already known since $SU(2)$ is compact.
The most direct approach to computing both the homology and the Euler characteristics would possibly be to find an explicit cellular decomposition of these groups. Again, my knowledge of basic Lie theory is failing me here: I don't see how to find such decompositions of the matrix groups.
Surely there are other methods of computing the homology which would exploit the Lie group structure, but I don't know of any.