Author: Olcay Coşkun

Title: Gluing Borel-Smith functions and the group of endo-trivial modules.

Abstract: The aim of this paper is to describe the group of endo-trivial modules for a $p$-group $P$, in terms of the obstruction
group for the gluing problem of Borel-Smith functions. Explicitly, we shall prove that there is a split exact sequence
\begin{displaymath}
\begin{diagram}
0&\rTo&\Zz &\rTo &T(P)&\rTo& \obs(C_b(P))&\rTo&0
\end{diagram}
\end{displaymath}
of abelian groups where $T(P)$ is the endo-trivial group of $P$, and $C_b(P)$ is the group of Borel-Smith functions on
$P$. As a consequence, we obtain a set of generators of the group $T(P)$ that coincides with the relative syzygies found
by Alperin. In order to prove the result, we solve gluing problems for the functor $B^*$ of super class functions, the
functor $\mathcal R_\Qq^*$ of rational class functions and the functor $C_b$ of Borel-Smith functions.

Status: Bulletin of London Mathematical Society (to appear)