Geometry & Topology

Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map

Matthew Ando, Andrew J Blumberg, and David Gepner

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We introduce a general theory of parametrized objects in the setting of –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.

Article information

Geom. Topol., Volume 22, Number 7 (2018), 3761-3825.

Received: 27 March 2015
Revised: 25 May 2017
Accepted: 20 July 2017
First available in Project Euclid: 14 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P99: None of the above, but in this section 55R70: Fibrewise topology

Thom spectrum parametrized spectra twisted Umkehr map


Ando, Matthew; Blumberg, Andrew J; Gepner, David. Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map. Geom. Topol. 22 (2018), no. 7, 3761--3825. doi:10.2140/gt.2018.22.3761.

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