# at.algebraic topology – Cohomology of BG, G non-connected Lie group, and spectral sequence relating to classifying space of connected component of the identity

Suppose $$G$$ is a Lie group, with $$pi_0(G)$$ not necessarily finite, but might as well assume $$G_0$$, the connected component of the identity, is compact.

In the case that $$pi_0(G)$$ is finite, then we know that there is an injection $$H^*(BG,mathbb{Q})to H^*(BG_0,mathbb{Q})$$, and this can apparently be seen via a spectral sequence argument, using the fact that the rational cohomology of $$Bpi_0(G)$$ is concentrated in degree zero. So this is some kind of Leray–Serre spectral sequence argument on either $$pi_0(G)to BG_0to BG$$ or $$BG_0to BGto Bpi_0(G)$$ (and I suspect the latter), probably using the degeneration and some kind of “edge homomorphism is injective” argument.

I suspect that in the case that we know something strong about the rational cohomology of $$Bpi_0(G)$$, then we might be able to say something in the case where $$pi_0(G)$$ is not finite.

Unfortunately my spectral sequence knowledge is limited, and I can’t find a treatment of spectral sequences that seems general enough to deal with this setup in general (namely non-simply-connected base, and possibly non-connected fibre, plus non-finiteness issues, depending on which fibration is used).

Is my intuition correct, that $$H^*(Bpi_0(G),mathbb{Q}) = H^0(Bpi_0(G),mathbb{Q})$$ can let us conclude something about how the cohomology of $$BG$$ relates to that of $$BG_0$$?

Also, what would be a good reference that covers a general-enough version of the relevant spectral sequence?