A Segal conjecture for p-completed classifying spaces

Details A Segal conjecture for p-completed classifying spaces A Segal conjecture for p-completed classifying spaces

2005-03-03

arxiv.org/abs/math/0503067v3

Mathematics

Algebraic Topology

31 pages. Minor changes. Final version, to appear in Advances in Mathematics

Scientific paper

10.1016/j.aim.2007.04.006

We formulate and prove a new variant of the Segal Conjecture describing the group of homotopy classes of stable maps from the p-completed classifying space of a finite group G to the classifying space of a compact Lie group K as the p-adic completion of the Grothendieck group of finite principal (G,K)-bundles whose isotropy groups are p-groups. Collecting the result for different primes p, we get a new and simple description of the group of homotopy classes of stable maps between (uncompleted) classifying spaces of groups. This description allows us to determine the kernel of the map from the Grothendieck group A(G,K) of finite principal (G,K)-bundles to the group of homotopy classes of stable maps from BG to BK.

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