On the relation of symplectic algebraic cobordism to hermitian K-theory

TL;DR

This paper demonstrates how hermitian K-theory can be reconstructed from algebraic symplectic cobordism within the motivic stable homotopy category, establishing an algebraic analogue of a classical topological theorem.

Contribution

It establishes a unique morphism between symplectic cobordism and hermitian K-theory spectra, linking their cohomology theories through a change of coefficients, thus providing an algebraic reconstruction of hermitian K-theory.

Findings

The morphism g : MSp -> BO is unique in the motivic stable homotopy category.

The induced morphism of cohomology theories aligns with a change of coefficients.

This work provides an algebraic analogue of Conner and Floyd's theorem on real K-theory reconstruction.

Abstract

We reconstruct hermitian K-theory via algebraic symplectic cobordism. In the motivic stable homotopy category SH(S) there is a unique morphism g : MSp -> BO of commutative ring T- spectra which sends the Thom class th^{MSp} to the Thom class th^{BO}. We show that the induced morphism of bigraded cohomology theories MSp^{*,*} -> BO^{*,*} is isomorphic to the morphism of bigraded cohomology theories obtained by applying to MSp^{*,*} the "change of (simply graded) coefficients rings" MSp^{4*,2*} -> BO^{4*,2*}. This is an algebraic version of the theorem of Conner and Floyd reconstructing real K-theory via symplectic cobordism.