  The main result of this paper is an application of the topology of the space
$Q(X)$ to obtain results for the cohomology of the symmetric group on $d$
letters, $\Sigma_d$, with `twisted' coefficients in various choices of Young
modules and to show that these computations reduce to certain natural questions
in representation theory. The authors extend classical methods for analyzing
the homology of certain spaces $Q(X)$ with mod-$p$ coefficients to describe the
homology $\HH_{\bullet}(\Sigma_d, V^{\otimes d})$ as a module for the general
linear group $GL(V)$ over an algebraically closed field $k$ of characteristic
$p$. As a direct application, these results provide a method of reducing the
computation of $\text{Ext}^{\bullet}_{\Sigma_{d}}(Y^{\lambda},Y^{\mu})$ (where
$Y^{\lambda}$, $Y^{\mu}$ are Young modules) to a representation theoretic
problem involving the determination of tensor products and decomposition
numbers. In particular, in characteristic two, for many $d$, a complete
determination of $\Hs Y^\lambda)$ can be found. This is the first nontrivial
class of symmetric group modules where a complete description of the cohomology
in all degrees can be given. For arbitrary $d$ the authors determine
$\HH^i(\Sigma_d,Y^\lambda)$ for $i=0,1,2$. An interesting phenomenon is
uncovered--namely a stability result reminiscent of generic cohomology for
algebraic groups. For each $i$ the cohomology $\HH^i(\Sigma_{p^ad},
Y^{p^a\lambda})$ stabilizes as $a$ increases. The methods in this paper are
also powerful enough to determine, for any $p$ and $\lambda$, precisely when
$\HH^{\bullet}(\sd,Y^\lambda)=0$. Such modules with vanishing cohomology are of
great interest in representation theory because their support varieties
constitute the representation theoretic nucleus.
