/sci/ - Science & Math

>>10727663
I'm trying to compute the homology of [math]K=K(G,1)[/math] using the path space fibration [math]\Omega K \rightarrow PK \rightarrow K[/math] and the Serre spectral sequence.

[math]\Omega K=K(G,0)[/math] is just a discrete set of [math]\# G[/math] points, so its only nontrivial homology group is [math]H_0(\Omega K)=\bigoplus_{\# G} \mathbb{Z}[/math]. So everything on the [math]E^2[/math] page is zero outside the [math]q=0[/math] row. I can get at least the first two groups there using Universal Coefficients and the Hurewicz map.

Now each diagonal [math]t=p+q[/math] on the [math]E^\infty[/math] page is supposed to give a filtration of the group [math]H_t(PK)[/math] whose homology I know since [math]PK[/math] is contractible. The problem is, the way the [math]E^2[/math] page is set up (pic related), there is no way for the groups in the [math]q=0[/math] row to vanish. This implies [math]\bigoplus_{\#G} \Z \cong \Z[/math] for instance, so something must have gone wrong here. Any thoughts/suggestions?