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Introduction to framed correspondences

Marc Hoyois and Nikolai Opdan

Vol. 6 (2025), 107–126
Abstract

We give an overview of the theory of framed correspondences in motivic homotopy theory. Motivic spaces with framed transfers are the analogue in motivic homotopy theory of E-spaces in classical homotopy theory, and in particular they provide an algebraic description of infinite 1-loop spaces. We will discuss the foundations of the theory (following Voevodsky, Garkusha, Panin, Ananyevskiy, and Neshitov), some applications such as the computations of the infinite loop spaces of the motivic sphere and of algebraic cobordism (following Elmanto, Hoyois, Khan, Sosnilo, and Yakerson), and some open problems.

Keywords
framed correspondences, motivic homotopy theory, infinite loop space, algebraic cobordisms
Mathematical Subject Classification
Primary: 14F42
Secondary: 55P47
Milestones
Received: 26 January 2022
Revised: 4 July 2022
Accepted: 20 July 2022
Published: 15 March 2025
Authors
Marc Hoyois
Fakultät für Mathematik
Universität Regensburg
Regensburg
Germany
Nikolai Opdan
Department of Mathematics
University of Oslo
Oslo
Norway