cohomology of the loop space of a Lie group: divided power structure

Question

Let $G$ be a complex simple simply connected algebraic group and $Gr_G$ be the corresponding affine Grassmannian (it is well known that $Gr_G$ is homotopy equivalent to the base loop space $\Omega K$ of the maximal compact subgroup $K\subset G$). Let $H=H^*(Gr,k)=H^*(\Omega K,k)$ be the cohomology ring where $k$ is a field of characteristic $p>0$. The computation of $H$ by Yun and Zhu in their paper "Integral homology of loop groups..." shows that the commutative $k$-algebra $H$ has a divided power structure, at least when $p$ is not too small. Can this be seen from first principles? In particular, is it related to Cartan's divided power structure on homology of a commutative simplicial $k$-algebra?

Answer 1

Let me make the following remarks regarding divided power structures and cohomology of loop spaces. Let us fix a field $\mathbb{F}$.

• 1) Le me recall the Eilenberg-Moore spectral sequence. Let us suppose that we have a fibration $$F\rightarrow E\rightarrow B$$ such that $B$ is $1$-connected, $H^k(B;\mathbb{F})$ and $H^k(E;\mathbb{F})$ are finite dimensional $\mathbb{F}$-vector spaces for any k. Then we have a convergent spectral sequence $$E_2^{p,q}=Tor^{H^*(B;\mathbb{F})}_{p,q}(H^*(E;\mathbb{F}),\mathbb{F})\implies H^*(F;\mathbb{F}).$$ In the case of the path fibration $$\Omega X\rightarrow PX\stackrel{ev_1}{\rightarrow} X$$ we get a convergent spectral sequence $$E_2^{p,q}=Tor^{H^*(X;\mathbb{F})}_{p,q}(\mathbb{F},\mathbb{F})\implies H^*(\Omega X;\mathbb{F}).$$

• 2) Behind the Eilenberg-Moore spectral sequence you have the Eilenberg-Moore quasi-isomorphism $$C^*(F;\mathbb{F})\simeq Tor^{C^*(B;\mathbb{F})}(C^*(E;\mathbb{F}),\mathbb{F})$$ in order to take into account of the multiplicative structure given by the cup product we have to be very careful and use the $E_{\infty}$-algebra structure on the singular cochains $C^*(-;\mathbb{F})$ and play with $E_{\infty}$-torsion products. This is very well explained in M. Mandell's paper "E_{\infty}-algebras and $p$-adic homotopy theory" (Topology, 2001, have a look at section $5$).

• 3) When one can replace the dg-algebra $C^*(X;\mathbb{F})$ by a commutative dg algebra i.e. when it is quasi-isomorphic to a cdga $A^*(X)$ the Torsion product $$Tor^{A^*(X)}(\mathbb{F};\mathbb{F})$$ is a cdga and by results of Cartan (Cartan's seminar 1954-1955, exposé $7$) when $char(\mathbb{F})>0$ its cohomology has a divided power algebra structure. The relationship between this divided power structure and the one coming from the homotopy ring of a simplicial commutative algebra is explained in B. Richter's paper "Divided power structures and chain complexes." Alpine perspectives on algebraic topology, 237–254, Contemp. Math., 504, Amer. Math. Soc., Providence, RI, 2009. The relationship is explained thanks to the Dold-kan correspondence and the classical Bar construction.

• 4) Last point, when can we replace $C^*(X;\mathbb{F})$ by a cdga? I know at least three cases: when $X$ is formal over the field $\mathbb{F}$, when the field is of characteristic zero, in the Anick's range if $X$ is $r$-connected and $dim(X)\leq r.char(\mathbb{F})$.

Conclusion: for any $1$-connected finite CW-complex $X$ the cohomology algebra of $\Omega X$ with coefficients in a field $\mathbb{F}$ has a divided power structure provided $char(\mathbb{F})>>0$. This divided power structure is inherited from the one coming from the classical Bar construction and is of simplicial nature.