Title : Gluing torsion endo-permutation modules
Authors : Serge Bouc and Jacques Thévenaz
Abstract :
Let $k$ be a field of characteristic $p$, and $P$ be a finite $p$-group. 
In this paper, we consider the problem of gluing compatible families of 
endo-permutation modules : being given a torsion element $M_Q$ in the Dade
group $D(N_P(Q)/Q)$, for each non-trivial subgroup $Q$ of $P$, subject to 
obvious compatibility conditions, we show that it is always possible to find 
an element $M$ in the Dade group of $P$ such that $\Defres_{N_P(Q)/Q}M=M_Q$ 
for all $Q$, but that $M$ need not be a torsion element of $D(P)$. The 
obstruction to this is controlled by an element in the zero-th cohomology 
group over $\F_2$ of the poset of elementary abelian subgroups of $P$ of 
rank at least 2. We also give an example of a similar situation, when $M_Q$ 
is only given for centric subgroups $Q$ of $P$.