# SL-oriented cohomology theories

## TL;DR

This paper establishes the uniqueness of SL^c-orientations in motivic cohomology theories with Zariski sheaf conditions, constructs Thom isomorphisms for SL^c-bundles, and explores ta-torsion characteristic classes.

## Contribution

It introduces a unique normalized SL^c-orientation for certain motivic cohomology theories and develops Thom isomorphisms in the SL-oriented setting.

## Key findings

- Uniqueness of SL^c-orientation under Zariski sheaf condition
- Construction of Thom isomorphisms for SL^c-bundles
- Euler class annihilated by the Hopf element in specific cases

## Abstract

We show that a representable motivic cohomology theory admits a unique normalized SL^c-orientation if the zeroth cohomology presheaf is a Zariski sheaf. We also construct Thom isomorphisms in SL-oriented cohomology for SL^c-bundles and obtain new results on the \eta-torsion characteristic classes, in particular, we prove that the Euler class of an oriented bundle admitting a (possibly non-orientable) odd rank subbundle is annihilated by the Hopf element.