On the cohomology ring of divided powers and free ∞-loop spaces

Given a pointed topological space $(X, \ast)$, its loop space is the space $\Omega(X, \ast)$ of continuous maps $\gamma \colon  [0,1] \to X$ such that $\gamma (0) = \gamma (1) = \ast$, suitably topologized. A $k$-fold loop space (or simply $k$-loop space) is the result of $k$ consecutive applications of the functor $\Omega$ to a pointed space, while an $\infty$-loop space is a topological space homotopically equivalent to a $k$-loop space for every $k$. $\infty$-loop spaces are extremely interesting for algebraic topologists.

 

There is a free functor $Q$ from the category of topological spaces to that of $\infty$-loop spaces. Given a space $X$, the ``building block'' of the object $Q(X)$ are the divided powers $D_kX = E(\mathcal{S}_n) \times_{\mathcal{S}_n} X^k$, which are also spaces of interest of their own.

 

In this talk, I will present a description of the cohomology ring of $Q(X)$ and $D_kX$ with coefficients in prime fields $\mathbb{F}_p$. This is joint work with prof. P. Salvatore and prof. D. Sinha.