On the Relation of Symplectic Algebraic Cobordism to Hermitian K-Theory

We reconstruct hermitian K-theory via algebraic symplectic cobordism. In the motivic stable homotopy category SH(S), there is a unique morphism phi: MSp -> BO of commutative ring T-spectra which sends the Thom class th(MSp) to the Thom class th(BO). Using phi we construct an isomorphism of bigraded ring cohomology theories on the category SmOp/S,phi over bar :MSp*,*(X,U)circle times MSp4*,0*(pt)BO4*,2*(pt) approximately equal to BO*,*(X,U). The result is an algebraic version of the theorem of Conner and Floyd reconstructing real K-theory using symplectic cobordism. Rewriting the bigrading as MSp(p,q) = MSp1q-p[q], we have an isomorphism phi over bar :MSp*[*](X,U)circle times MSp0[2*](pt)KO0[2*](pt) approximately equal to KO*[*](X,U), where the KOi[n](X,U) are Schlichting's hermitian K-theory groups.