Vol. 301, No. 2, 2019

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K-theory of affine actions

James Waldron

Vol. 301 (2019), No. 2, 639–666
DOI: 10.2140/pjm.2019.301.639

For a Lie group G and a vector bundle E we study those actions of the Lie group TG on E for which the action map TG × E E is a morphism of vector bundles, and call those affine actions. We prove that the category VectTG aff(X) of such actions over a fixed G-manifold X is equivalent to a certain slice category gXVectG(X). We show that there is a monadic adjunction relating VectTGaff(X) to VectG(X), and the right adjoint of this adjunction induces an isomorphism of Grothendieck groups KTGaff(X)KOG(X). Complexification produces analogous results involving TG and KG(X).

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vector bundles, equivariant K-theory, differential geometry
Mathematical Subject Classification 2010
Primary: 19E99
Received: 23 October 2017
Accepted: 28 December 2018
Published: 24 October 2019
James Waldron
School of Mathematics, Statistics and Physics
Newcastle University
Newcastle upon Tyne
United Kingdom