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Segal's Spectral Sequence in Twisted Equivariant K-theory for Proper and Discrete Actions

Part of: Homology and cohomology theories Homological algebra Topological $K$-theory

Published online by Cambridge University Press:  23 January 2018

Noé Bárcenas ,
Jesús Espinoza ,
Bernardo Uribe  and
Mario Velásquez
Noé Bárcenas*
Affiliation:
Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Apartado Postal 61-3 Xangari, Morelia, Michoacán 58089, Mexico (barcenas@matmor.unam.mx; jespinoza@matmor.unam.mx)
Jesús Espinoza
Affiliation:
Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Apartado Postal 61-3 Xangari, Morelia, Michoacán 58089, Mexico (barcenas@matmor.unam.mx; jespinoza@matmor.unam.mx)
Bernardo Uribe
Affiliation:
Departamento de Matemáticas y Estadística, Universidad del Norte, Km 5 Vía Puerto Colombia, Barranquilla, Colombia (bjongbloed@uninorte.edu.co)
Mario Velásquez
Affiliation:
Departamento de Matemáticas, Pontificia Universidad Javeriana, Cra. 7 No. 43-82 – Edificio Carlos Ortíz, 5to piso, Bogotá D.C., Colombia (mario.velasquez@javeriana.edu.co)
*
*Corresponding author.
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Abstract

We use a spectral sequence developed by Graeme Segal in order to understand the twisted G-equivariant K-theory for proper and discrete actions. We show that the second page of this spectral sequence is isomorphic to a version of Bredon cohomology with local coefficients in twisted representations. We furthermore explain some phenomena concerning the third differential of the spectral sequence, and recover known results when the twisting comes from finite order elements in discrete torsion.

Keywords

twisted equivariant K-theoryBredon cohomologyproper actionstwisted representations

MSC classification

Primary: 19L47: Equivariant $K$-theory 19L50: Twisted $K$-theory; differential $K$-theory 55N91: Equivariant homology and cohomology
Secondary: 18G40: Spectral sequences, hypercohomology
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 1 , February 2018 , pp. 121 - 150
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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