The Atiyah--Segal completion theorem in twisted K-theory

Details The Atiyah--Segal completion theorem in twisted K-theory The Atiyah--Segal completion theorem in twisted K-theory

2008-09-08

arxiv.org/abs/0809.1273v1

Mathematics

Algebraic Topology

16 pages

Scientific paper

In this note we prove the analogue of the Atiyah-Segal completion theorem for equivariant twisted K-theory in the setting of an arbitrary compact Lie group G and an arbitrary twisting of the usually considered type. The theorem generalizes a result by C. Dwyer, who has proven the theorem for finite G and twistings of a more restricted type. While versions of the general result have been known to experts, to our knowledge no proof appears in the current literature. Our goal is to fill in this gap. The proof we give proceeds in two stages. We first prove the theorem in the case of a twisting arising from a graded central extension of G, following the Adams-Haeberly-Jackowski-May proof of the classical Atiyah-Segal completion theorem. After establishing that the theorem holds for this special class of twistings, we then deduce the general theorem by a Mayer-Vietoris argument.

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