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 $n gg 0$.
A p-local compact group is an algebraic object modelled on the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups. In the study of these objects unstable Adams operations, are of fundamental importance. In this p
aper we define unstable Adams operations within the theory of p-local compact groups, and show that such operations exist under rather mild conditions. More precisely, we prove that for a given p-local compact group G and a sufficiently large positive integer $m$, there exists an injective group homomorphism from the group of p-adic units which are congruent to 1 modulo p^m to the group of unstable Adams operations on G
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.
Given an operad A of topological spaces, we consider A-monads in a topological category C . When A is an A-infinity-operad, any A-monad K : C -> C can be thought of as a monad up to coherent homotopies. We define the completion functor with respect t
o an A-infinity-monad and prove that it is an A-infinity-monad itself.
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.
We construct a combinatorial model of an A-infinity-operad which acts simplicially on the cobar resolution (not just its total space) of a simplicial set with respect to a ring R.
