HOMOLOGY DECOMPOSITIONS FOR CLASSIFYING SPACES OF 
FINITE GROUPS ASSOCIATED TO MODULAR REPRESENTATIONS
  
                               D. Notbohm
For a prime $p$, a homology decomposition of a the classifying 
space $BG$ of a finite group $G$ consist of a functor 
$F :\BD >>> \spaces$ from a small category into the category 
of spaces and a map $\hcl{} F >>> BG$ from 
the homotopy colimit to $BG$ which induces an isomorphism in 
mod-$p$ homology. Associated to a modular representation 
$G >>> Gl(n;\fp)$ we construct a family 
of subgroups closed under conjugation, which gives rise to 
three different homology decompositions, the so called subgroup, 
centralizer and normalizer decomposition. For an action of 
$G$ on a $\fp$-vector space $V$, the collection consist of the 
isotropy groups of all nontrivial proper subspaces of $V$ with 
nontrivial $p$-Sylow subgroup. These decomposition formulas 
connect the modular representation theory of $G$ with the homotopy 
theory of $BG$.