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MULTIPLICATIVE PARAMETRIZED HOMOTOPY THEORY VIA SYMMETRIC SPECTRA IN RETRACTIVE SPACES

Part of: Homotopy theory

Published online by Cambridge University Press:  19 March 2020

FABIAN HEBESTREIT ,
STEFFEN SAGAVE  and
CHRISTIAN SCHLICHTKRULL
FABIAN HEBESTREIT
Affiliation:
Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115Bonn, Germany; [email protected]
STEFFEN SAGAVE
Affiliation:
IMAPP, Radboud University Nijmegen, PO Box 9010, 6500GL Nijmegen, The Netherlands; [email protected]
CHRISTIAN SCHLICHTKRULL
Affiliation:
Department of Mathematics, University of Bergen, P.O. Box 7803, 5020Bergen, Norway; [email protected]

Abstract

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In order to treat multiplicative phenomena in twisted (co)homology, we introduce a new point-set-level framework for parametrized homotopy theory. We provide a convolution smash product that descends to the corresponding $\infty$-categorical product and allows for convenient constructions of commutative parametrized ring spectra. As an immediate application, we compare various models for generalized Thom spectra. In a companion paper, this approach is used to compare homotopical and operator algebraic models for twisted $K$-theory.

MSC classification

Primary: 55P43: Spectra with additional structure ($E_infty$, $A_infty$, ring spectra, etc.)
Secondary: 55P42: Stable homotopy theory, spectra
Type
Topology
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e16
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (http://creativecommons.org/licenses/by-nc-sa/4.0/), which permits noncommercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
Copyright
© The Author(s) 2020

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