On the relation of special linear algebraic cobordism to Witt groups

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.