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

Abstract

We reconstruct hermitian K-theory via algebraic symplectic cobordism. In the motivic stable homotopy category SH(S), there is a unique morphism ϕ: MSp → BO of commutative ring T-spectra which sends the Thom class th^MSp to the Thom class th^BO. Using ϕ we construct an isomorphism of bigraded ring cohomology theories on the category \({\mathop{\rm Sm}\nolimits} {\mathcal O}p/S,\bar \varphi :{{\mathop{\rm MSp}\nolimits} ^{*,*}}(X,U){ \otimes _{{\rm{MS}}{{\rm{p}}^{4*,0*}}({\rm{pt}})}}{\rm{B}}{{\rm{O}}^{4*,2*}}({\rm{pt}}) \cong {\rm{B}}{{\rm{O}}^{*,*}}(X,U)\). 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 = MSp_1q−p^[q], we have an isomorphism \(\bar \varphi :{{\mathop{\rm MSp}\nolimits} _*}^{[*]}(X,U){ \otimes _{{\rm{MSp}}_0^{[2*]}({\rm{pt}})}}{\rm{KO}}_0^{[2*]}({\rm{pt}}) \cong {\rm{K}}{{\rm{O}}_*}^{[*]}(X,U)\), where the KO_i^[n](X,U) are Schlichting’s hermitian K-theory groups.