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

I. A. Panin ; C. Walter

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 180-192

Résumé

We reconstruct hermitian $K$-theory via algebraic symplectic cobordism. In the motivic stable homotopy category $\mathrm {SH}(S)$, there is a unique morphism $\varphi \colon \mathbf {MSp}\to \mathbf {BO}$ of commutative ring $T$-spectra which sends the Thom class $\mathrm {th}^{\mathbf {MSp}}$ to the Thom class $\mathrm {th}^{\mathbf {BO}}$. Using $\varphi $ we construct an isomorphism of bigraded ring cohomology theories on the category $\mathcal Sm\mathcal Op/S$, $\overline \varphi \colon \mathbf {MSp}^{*,*}(X,U)\otimes _{\mathbf {MSp}^{4*,2*}(\mathrm {pt})} \mathbf {BO}^{4*,2*}(\mathrm {pt}) \cong \mathbf {BO}^{*,*}(X,U)$. The result is an algebraic version of the theorem of Conner and Floyd reconstructing real $K$-theory using symplectic cobordism. Rewriting the bigrading as $\mathbf {MSp}^{p,q}=\mathbf {MSp}^{[q]}_{2\smash {q-p}}$, we have an isomorphism $\overline \varphi \colon \mathbf {MSp}^{[*]}_*(X,U)\otimes _{\mathbf {MSp}^{[2*]}_0(\mathrm {pt})} \mathrm {KO}^{[2*]}_0(\mathrm {pt}) \cong \mathrm {KO}^{[*]}_*(X,U)$, where the $\mathrm {KO}^{[n]}_i(X,U)$ are Schlichting's hermitian $K$-theory groups.