Framed transfers and motivic fundamental classes

TL;DR

This paper explores the relationship between different transfer theories in motivic homotopy theory, introducing a new category of finite E-correspondences and connecting framed transfers with motivic fundamental classes.

Contribution

It compares two types of transfers in motivic cohomology, introduces finite E-correspondences, and shows how framed correspondences relate to E-module spectra via fundamental classes.

Findings

Comparison of framed and Gysin transfers in motivic cohomology

Introduction of finite E-correspondences for motivic spectra

Factorization of framed correspondences through finite E-correspondences

Abstract

We relate the recognition principle for infinite $\mathbf P^1$-loop spaces to the theory of motivic fundamental classes of D\'eglise, Jin, and Khan. We first compare two kinds of transfers that are naturally defined on cohomology theories represented by motivic spectra: the framed transfers given by the recognition principle, which arise from Voevodsky's computation of the Nisnevish sheaf associated with $\mathbf A^n/(\mathbf A^n-0)$, and the Gysin transfers defined via Verdier's deformation to the normal cone. We then introduce the category of finite E-correspondences for E a motivic ring spectrum, generalizing Voevodsky's category of finite correspondences and Calm\`es and Fasel's category of finite Milnor-Witt correspondences. Using the formalism of fundamental classes, we show that the natural functor from the category of framed correspondences to the category of E-module spectra factors through the category of finite E-correspondences.