User:Ecpeterson/Bar spectral sequence

Start with a homology theory $$E$$ and an $$H$$-group $$G$$ compatible with $$E$$ in that we have a Künneth-like isomorphism $$E_* G^{\wedge r} \cong E_* G^{\otimes r}$$. Take the model for the classifying space $$BG$$ to be $$\coprod_n \Delta^n \times G^n / \sim$$, where $$\sim$$ identifies faces of $$G^n \times \Delta^n$$ with elements of $$G^{n-1} \times \Delta^{n-1}$$. Then there's an increasing, based filtration $$B_r G = \coprod_{n \le r} \Delta^n \times G^n / \sim$$ with $$BG = \mathrm{colim}_r\;B_r G$$ and $$B_0 G = \mathrm{pt}$$, giving rise to a spectral sequence from a filtered complex converging to $$E_* BG$$. The filtration quotients are $$F_r = B_r G / B_{r-1} G = \Sigma^r G^r$$, and the induced differential on the $$E^1$$ page is the bar differential. Hence, the $$E^2$$ page of the spectral sequence coincides with $$H_{*, *} E_* G = Tor^{E_* G}_{*, *}(E_*, E_*)$$.

Naturality w.r.t Hopf ring structure.

Applications in topology: Nilpotence I, Ravenel-Wilson on the Morava $$K$$-theory of Eilenberg-Mac Lane spaces, ...?