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

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Lahtinen, Anssi

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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.

Comment: 13 pages. The numbering of theorems, lemmas and corollaries is the same as in the published version

Comment: 13 pages. The numbering of theorems, lemmas and corollaries is the same as in the published version

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Mathematics - Algebraic Topology, Mathematics - K-Theory and Homology, 55N15, 19L50, 19L47, 55P91