Introduction to Framed Correspondences

TL;DR

This paper provides an overview of framed correspondences in motivic homotopy theory, highlighting their foundational aspects, applications like computing infinite loop spaces, and open problems in the field.

Contribution

It summarizes the development and applications of framed correspondences, connecting motivic spaces with classical homotopy concepts and outlining recent advances.

Findings

Framework for motivic spaces with framed transfers

Calculations of infinite loop spaces of motivic sphere and algebraic cobordism

Discussion of open problems in framed correspondences

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 $E_{\infty}$-spaces in classical homotopy theory, and in particular they provide an algebraic description of infinite $\mathbb{P}^1$-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.