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Fibrant resolutions for motivic Thom spectra

Grigory Garkusha and Alexander Neshitov

Vol. 8 (2023), No. 3, 421–488
Abstract

Using the theory of framed correspondences developed by Voevodsky and the machinery of framed motives introduced and developed in recent work of Garkusha and Panin, various explicit fibrant resolutions for a motivic Thom spectrum E are constructed in this paper. The bispectrum

ME𝔾(X) = (M E(X),ME(X)(1),ME(X)(2),),

each term of which is a twisted E-framed motive of X, is introduced and shown to represent X+ E in the category of bispectra. As a topological application, it is proved that the E-framed motive with finite coefficients ME(pt )(pt )N, N > 0, of the point pt = Spec k evaluated at pt is a quasifibrant model of the topological S2-spectrum Re 𝜖(E)N whenever the base field k is algebraically closed of characteristic zero with an embedding 𝜀 : k. Furthermore, the algebraic cobordism spectrum MGL is computed in terms of Ω-correspondences. It is also proved that MGL is represented by a bispectrum each term of which is a sequential colimit of simplicial smooth quasiprojective varieties.

Keywords
motivic homotopy theory, motivic Thom spectra, $E$-framed motives
Mathematical Subject Classification
Primary: 14F42, 55P42
Milestones
Received: 18 October 2022
Revised: 18 June 2023
Accepted: 3 August 2023
Published: 27 August 2023
Authors
Grigory Garkusha
Department of Mathematics
Swansea University
Swansea
United Kingdom
Alexander Neshitov
Department of Mathematics
Western University
London, ON
Canada