Orbifold bordism and duality for finite orbispectra

Pardon, John

doi:10.2140/gt.2023.27.1747

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Orbifold bordism and duality for finite orbispectra

John Pardon

Geometry & Topology 27 (2023) 1747–1844

DOI: 10.2140/gt.2023.27.1747

Abstract

We construct the stable (representable) homotopy category of finite orbispectra, whose objects are formal desuspensions of finite orbi-CW–pairs by vector bundles and whose morphisms are stable homotopy classes of (representable) relative maps. The stable representable homotopy category of finite orbispectra admits a contravariant involution extending Spanier–Whitehead duality. This duality relates homotopical cobordism theories (cohomology theories on finite orbispectra) represented by global Thom spectra to geometric (derived) orbifold bordism groups (homology theories on finite orbispectra). This isomorphism extends the classical Pontryagin–Thom isomorphism and its known equivariant generalizations.

Keywords

orbispaces, orbispectra, Spanier–Whitehead duality, global homotopy theory, Pontryagin–Thom construction, stable homotopy theory, orbifold bordism, equivariant bordism

Mathematical Subject Classification

Primary: 55M05, 55P25, 55P42, 55P91, 57R85

Secondary: 55N91, 55Q91, 55R91, 55U30, 57R91

References

Publication

Received: 30 July 2020

Revised: 16 August 2021

Accepted: 6 January 2022

Published: 27 July 2023

Proposed: Stefan Schwede

Seconded: Jesper Grodal, Martin R Bridson

Authors

John Pardon
Department of Mathematics Princeton University Princeton, NJ United States
https://web.math.princeton.edu/~jpardon/

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Volume 27, issue 5 (2023)