Gluing torsion endo-permutation modules

Let k be a field of characteristic p, and P be a finite p-group, where p is an odd prime. In this paper, we consider the problem of gluing compatible families of endo-permutation modules : being given a torsion element MQ in the Dade group D(NP (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 DefresPNP (Q)/Q M = MQ 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 F2 of the poset of elementary abelian subgroups of P of rank at least 2. We also give an example of a similar situation, when MQ is only given for centric subgroups Q of P . Moreover, general results about biset functors and the Dade functor are given in two appendices. AMS Subject classification : 20C20