# The unit map of the algebraic special linear cobordism spectrum

## TL;DR

This paper computes the unit map of the algebraic special linear cobordism spectrum, showing it induces an isomorphism on certain homotopy sheaves by explicitly describing motivic Thom spectra using framed correspondences.

## Contribution

It provides an explicit description of motivic Thom spectra's homotopy groups and proves the unit map induces an isomorphism on $ ext{G}_m$-homotopy sheaves.

## Key findings

- Explicit description of non-negative $ ext{G}_m$-homotopy groups of motivic Thom spectra.
- The unit map of the algebraic special linear cobordism spectrum induces an isomorphism on $ ext{G}_m$-homotopy sheaves.

## Abstract

In joint work with Elmanto, Hoyois, Khan and Sosnilo, we computed infinite $\mathbb{P}^1$-loop spaces of motivic Thom spectra, using the technique of framed correspondences. This result allows us to express non-negative $\mathbb{G}_m$-homotopy groups of motivic Thom spectra in terms of geometric generators and relations. Using this explicit description, we show that the unit map of the algebraic special linear cobordism spectrum induces an isomorphism on $\mathbb{G}_m$-homotopy sheaves.