Homology, Homotopy and Applications

Volume 22 (2020)

Number 2

Topological $K$-theory of equivariant singularity categories

Pages: 1 – 29

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n2.a1


Michael K. Brown (Department of Mathematics, University of Wisconsin, Madison, Wisc., U.S.A.)

Tobias Dyckerhoff (Fachbereich Mathematik, Universität Hamburg, Germany)


We study the topological $K$-theory spectrum of the $\operatorname{dg}$ singularity category associated to a weighted projective complete intersection. We calculate the topological $K$-theory of the $\operatorname{dg}$ singularity category of a weighted projective hypersurface in terms of its affine Milnor fiber and monodromy operator, and, as an application, we obtain a lift of the Atiyah–Bott–Shapiro construction to the level of spectra.


$\operatorname{dg}$-category, matrix factorization, Milnor fiber, topological $K$-theory

2010 Mathematics Subject Classification

14A22, 18D20, 19D55, 19L47, 32S55

Received 6 March 2019

Received revised 7 August 2019

Accepted 7 August 2019

Published 26 February 2020