Nicholas J. Kuhn, 

New cohomological relationships among loopspaces, symmetric products, 
and Eilenberg-MacLane spaces

\begin{abstract}
Let $T(j)$ be the  dual of the $j^{th}$ Brown-Gitler spectrum 
(at the prime 2) with top class in dimension $j$. Then it is 
known that $T(j)$ is a retract of a suspension spectrum, is 
dual to a stable summand of $\Omega^2 S^3$, and that the homotopy 
colimit of a certain sequence $T(j) \rightarrow T(2j) \rightarrow 
\ldots$ is a wedge of stable summands of $K(V,1)$'s, where $V$ 
denotes an elementary abelian 2 group. In particular, when one 
starts with $T(1)$, one gets $K(Z/2,1) = RP^{\infty}$ as one of 
the summands. 

Refining a question posed by Doug Ravenel, I discuss a generalization 
of this picture.  I consider certain finite spectra $T(n,j)$ 
for $n,j \geq 0$ (with $T(1,j) = T(j)$), dual to summands of 
$\Omega^{n+1}S^{N}$, conjecture generalizations of all of the 
above, and prove that all these conjectures are correct in 
cohomology.  So, for example, $T(n,j)$ has unstable cohomology, 
and the cohomology of the colimit of a certain sequence $T(n,j) 
\rightarrow T(n,2j) \rightarrow \dots$ agrees with the cohomology 
of the wedge of stable summands of $K(V,n)$'s corresponding to the 
wedge occurring in the $n=1$ case above.  

One can also map the $T(n,j)$ to each other as $n$ varies, and the 
cohomological calculations suggest conjectures related to symmetric 
products of spheres. 
\end{abstract}