Volume 22, issue 7 (2018)

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Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map

Matthew Ando, Andrew J Blumberg and David Gepner

Geometry & Topology 22 (2018) 3761–3825

We introduce a general theory of parametrized objects in the setting of ß–categories. Although parametrised spaces and spectra are the most familiar examples, we establish our theory in the generality of families of objects of a presentable –category parametrized over objects of an –topos. We obtain a coherent functor formalism describing the relationship of the various adjoint functors associated to base-change and symmetric monoidal structures.

Our main applications are to the study of generalized Thom spectra. We obtain fiberwise constructions of twisted Umkehr maps for twisted generalized cohomology theories using a geometric fiberwise construction of Atiyah duality. In order to characterize the algebraic structures on generalized Thom spectra and twisted (co)homology, we express the generalized Thom spectrum as a categorification of the well-known adjunction between units and group rings.

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Thom spectrum, parametrized spectra, twisted Umkehr map
Mathematical Subject Classification 2010
Primary: 55P99, 55R70
Received: 27 March 2015
Revised: 25 May 2017
Accepted: 20 July 2017
Published: 6 December 2018
Proposed: Mark Behrens
Seconded: Ralph Cohen, Peter Teichner
Matthew Ando
Department of Mathematics
The University of Illinois at Urbana-Champaign
Urbana, IL
United States
Andrew J Blumberg
Department of Mathematics
University of Texas at Austin
Austin, TX
United States
David Gepner
Department of Mathematics
Purdue University
West Lafayette, IN
United States