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Orbifold bordism and duality for finite orbispectra

John Pardon

Geometry & Topology 27 (2023) 1747–1844
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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