Let $S(V)$ be a complex linear sphere of a finite group $G$. %the space of unit vectors in a complex representation $V$ of a finite group $G$. Let $S(V)^{*n}$ denote the $n$-fold join of $S(V)$ with itself and let $aut_G(S(V)^*)$ denote the space of
$G$-equivariant self homotopy equivalences of $S(V)^{*n}$. We show that for any $k geq 1$ there exists $M>0$ which depends only on $V$ such that $|pi_k aut_G(S(V)^{*n})| leq M$ is for all $n gg 0$.
We relate the construction of groups which realize saturated fusion systems and signaliser functors with homology decompositions of p-local finite groups. We prove that the cohomology ring of Robinsons construction is in some precise sense very close
to the cohomology ring of the fusion system it realizes.
We construct an analogue of the normaliser decomposition for p-local finite groups (S,F,L) with respect to collections of F-centric subgroups and collections of elementary abelian subgroups of S. This enables us to describe the classifying space of a
p-local finite group, before p-completion, as the homotopy colimit of a diagram of classifying spaces of finite groups whose shape is a poset and all maps are induced by group monomorphisms.
