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}