The localization of orthogonal calculus with respect to homology

Taggart, Niall

doi:10.2140/agt.2024.24.1505

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The localization of orthogonal calculus with respect to homology

Niall Taggart

Algebraic & Geometric Topology 24 (2024) 1505–1549

DOI: 10.2140/agt.2024.24.1505

Abstract

For a set of maps of based spaces $S$ we construct a version of Weiss’s orthogonal calculus which depends only on the $S$ –local homotopy type of the functor involved. We show that $S$ –local homogeneous functors of degree $n$ are equivalent to levelwise $S$ –local spectra with an action of the orthogonal group $O (n)$ via a zigzag of Quillen equivalences between appropriate model categories. Our theory specialises to homological localizations and nullifications at a based space. We give a variety of applications including a reformulation of the telescope conjecture in terms of our local orthogonal calculus and a calculus version of Postnikov sections. Our results also apply when considering the orthogonal calculus for functors which take values in spectra.

Keywords

orthogonal calculus, Bousfield localization, homological localization, nullification, calculus of functors

Mathematical Subject Classification

Primary: 55P60, 55P65

Secondary: 55N20, 55P42

References

Publication

Received: 14 February 2022

Revised: 11 October 2022

Accepted: 20 November 2022

Published: 28 June 2024

Authors

Niall Taggart
Mathematical Institute Utrecht University Utrecht Netherlands

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Volume 24, issue 3 (2024)