On spaces of self homotopy equivalences of p-completed classifying
spaces of finite groups and homotopy group extensions
By C. Broto and R. Levi
AMS Classification: 55R35
Address:
Carles Broto, Departament de Mathematiques, Universitat
Autonoma de Barcelona, E08193 Bellaterra, Spain
Ran Levi, Department of Mathematical Sciences, University of
Aberdeen, Aberdeen, AB24 3UE, Scotland
Email:
Carles Broto
Ran Levi
Fix a prime p. A mod-p homotopy group extension of a group $\pi$ by
a group G is a fibration with base space $B\pi^\wedge_p$ and fibre
$BG^\wedge_p$. In this paper we study homotopy group extensions for
finite groups. We observe that there is a strong analogy between homotopy
group extensions and ordinary group extensions. The study involves
investigating the space of self homotopy equivalences of a
p-completed classifying space. In particular we show that under
the appropriate assumption on $G$, the identity component of this
space is homotopy equivalent to $BZ(G)$, the classifying space of
the centre of $G$. We proceed by studying the group of components.
We show that this group maps into a group of natural equivalences
of a certain functor with kernel and cokernel, which are
computable in terms of the first and second derived functors of
the inverse limit for a certain diagram of abelian groups.