On the relation of special linear algebraic cobordism to Witt groups

TL;DR

This paper establishes a deep connection between special linear algebraic cobordism and derived Witt groups, showing an isomorphism that bridges cobordism theories and Witt groups for smooth varieties.

Contribution

It reconstructs derived Witt groups via special linear algebraic cobordism and proves an isomorphism linking their cohomology theories, extending previous hermitian K-theory results.

Findings

An isomorphism between MSL^{*,*}(X)[h^{-1}] and Laurent polynomial ring over W^*(X).

The morphism of ring cohomology theories maps Thom classes appropriately.

The result generalizes Panin and Walter's reconstruction of hermitian K-theory.

Abstract

We reconstruct derived Witt groups via special linear algebraic cobordism. There is a morphism of ring cohomology theories which sends the canonical Thom class in special linear cobordism to the Thom class in the derived Witt groups. We show that for every smooth variety X this morphism induces an isomorphism between MSL^{*,*}(X)[h^{-1}] with the "extended" coefficient ring MSL^{4*,2*}(pt) -> W^{2*}(pt) and Laurent polynomial ring over the derived Witt groups W^*(X), where h is the stable Hopf map. This result is an analogue of the result by Panin and Walter reconstructing hermitian K-theory using symplectic algebraic cobordism.