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).

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