MIT Topology Seminar
Monday, November 7, 2005
Room 2-142, 4:30pm
Kari Ragnarsson will speak on:
A Segal conjecture for p-completed classifying spaces
Abstract: As was predicted by Adams and Miller, and shown by
Lewis-May-McClure, one consequence of Carlsson's solution of the Segal
conjecture is the description of the group {BG,BH} of homotopy classes of
stable maps between classifying spaces of finite groups G and H as the
completion of A(G,H) at the augmentation ideal I(G) of the Burnside ring
A(G). (Here A(G,H) denotes the Grothendieck group completion of the monoid
of isomorphism classes of finite (G x H)-sets such that the induced
H-action is free, and A(G) can be regarded as the special case where H is
the trivial group.) Unfortunately such completions are very difficult to
calculate in general. However, Lewis-May showed that in the special case
where G is a p-group, I(G)-adic completion agrees with p-adic completion.
In this talk I will illustrate how the simplification of Lewis-May can be
extended to general finite groups G, but at the cost of p-completing
classifying spaces. More precisely, let A_p(G,H) denote the submodule of
A(G,H) generated by those (G x H)-sets whose isotropy groups under the
G-action are p-groups. I will show that {BG^_p,BH} is the p-completion of
A_p(G,H). Since BG is the wedge sum of the BG^_p for different primes
p, we can collect these results to get a new and simple description of
{BG,BH}.