Geometry, Topology and Dynamics Seminar
Kari Ragnarsson
Northwestern University
Homotopy classifications of p-completed classifying spaces.
Abstract: In algebraic topology one typically applies powerful algebraic
invariants to encode homotopy properties of topological spaces. In certain
cases it is possible and useful to reverse this process by assigning a
space to an algebraic object. An instance of this is the assignment to a
finite group G of a classifying space BG, whence the group G can be
recovered as the fundamental group. Furthermore, group homomorphisms
between finite groups correspond bijectively to homotopy classes of maps
between their classifying spaces.
In this talk I will discuss how this correspondence changes when we focus
on properties relative to a prime p. Topologically this means applying the
p-completion functor to BG. I will present three classification theorems
for p-completed classifying spaces of finite groups.
First, the unstable classification, predicted by Martino-Priddy and proved
by Oliver, which classifies the homotopy type of the p-completed
classifying space of G via the fusion system of G.
Second, the stable classification, due to Martino-Priddy, which classifies
the stable homotopy type of the p-completed classifying space of G via
weaker data, which loosely speaking can be regarded as a linearisation of
the fusion system.
Finally, the partially stable classification, which links the unstable and
stable classifications. This is the surprising result that, by keeping
track of inclusions of Sylow subgroups, the stable homotopy type of the
p-completed classifying space of G can again be classified via the fusion
data of G. This classification also gives a simple description of maps
realising stable homotopy equivalences (while preserving the inclusions
of Sylow subgroups).
Monday October 30, 2006 at 3:00 PM in SEO 512