Gluing torsion endo-permutation modules

Let k be a field of characteristic p, and let P be a finitep-group, where p is an odd prime. In this paper, we considerthe problem of gluing compatible families of endo-permutationmodules: being given a torsion element M_Q in the Dade groupD(N_P(Q)/Q), for each non-trivial subgroup Q of P, subject toobvious compatibility conditions, we show that it is alwayspossible to find an element M in the Dade group of P such that for all Q, but that Mneed not be a torsion element of D(P). The obstruction to thisis controlled by an element in the zeroth cohomology group over₂ of the poset of elementary abelian subgroups of P of rankat least 2. We also give an example of a similar situation,when M_Q is only given for centric subgroups Q of P. Moreover,general results about biset functors and the Dade functor aregiven in two appendices.