Title : Borel-Smith functions and the Dade group.

Authors : Serge Bouc and Ergün Yalçin

Abstract :
We show that there is an exact sequence of biset functors over
$p$-groups 
$$0 \to C_b \maprt{j} B^* \maprt{\Psi} D^{\Omega } \to 0$$ 
where $C_b$ is the biset functor for the group of Borel-Smith
functions, $B^*$ is the dual of Burnside ring functor,  $D^{\Omega}$
is the functor for the subgroup of Dade group generated by relative
syzygies, and the natural transformation $\Psi$ is the
transformation recently introduced by the first author in
\cite{Bouc-A remark on Dade and Burnside}. We also show that the
kernel of mod $2$ reduction of $\Psi $ is naturally equivalent to
the unit group of Burnside ring functor $B^{\times}$ and obtain
exact sequences involving the torsion part of $D^{\Omega}$, mod $2$
reduction of $C_b$, and the unit group of Burnside ring
functor~$B^{\times}$.