Title :  
Gluing endo-permutation modules

Abstract :
In this paper, I show that if p is an odd prime, and if P is a finite p-group, then there exists an exact sequence of abelian groups
0--> T(P)-->D(P)--> lim(P)--> H^1(a(P),Z)^(P)
where D(P) is the Dade group of P and T(P) is the subgroup of endo-trivial modules.
Here lim(P) is the group of sequences of compatible elements in the Dade groups D(N_P(Q)/Q) for non trivial subgroups Q of P. The poset a(P) is the set of elementary abelian subgroups of rank at least 2 of P, ordered by inclusion. The group $H^1(a(P),Z)^(P) is the subgroup of H^1(a(P),Z) consisting of classes of P-invariant 1-cocycles.
A key result to prove that the above sequence is exact is a characterization of elements of 2D(P) by sequences of integers, indexed by sections (T,S) of P such that T/S is elementary abelian of rank 2, fulfilling certain conditions associated to subquotients of P which are either elementary abelian of rank 3, or extraspecial of order p^3 and exponent p.

Status : 
To appear in the Journal of Group Theory