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On the relation of symplectic algebraic cobordism to Hermitian \(K\)-theory. (English. Russian original) Zbl 1442.19012
Proc. Steklov Inst. Math. 307, 162-173 (2019); translation from Tr. Mat. Inst. Steklova 307, 180-192 (2019).
Summary: We reconstruct hermitian \(K\)-theory via algebraic symplectic cobordism. In the motivic stable homotopy category \(SH(S)\), there is a unique morphism \(\varphi : \mathbf{MSp} \rightarrow \mathbf{BO}\) 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 \(\mathcal{S}m\mathcal{O}p/S\), \(\bar{\varphi} : \mathbf{MSp}^{*,*}(X,U)\otimes_{\mathbf{MSp}^{4*,0*}(\mathrm{pt})} \mathrm{BO}^{4*,2*}(\mathrm{pt}) \cong \mathrm{BO}^{*,*}(X,U)\). The result is an algebraic version of the theorem of Conner and Floyd reconstructing real \(K\)-theory using symplectic cobordism. Rewriting the bigrading as \(\mathbf{MSp}^{ p,q } = \mathbf{MSp}_{1 q - p }^{[q]} \), we have an isomorphism \(\bar{\varphi}:\mathbf{MSp}_*^{[*]}(X,U) \otimes_{\mathbf{MSp}_0^{[2*]}(\mathrm{pt})} \mathrm{KO}_0^{[2*]}(\mathrm{pt}) \cong \mathrm{KO}_*^{[*]}(X,U)\), where the \(KO_i^{[n]} (X,U)\) are Schlichting’s hermitian \(K\)-theory groups.
MSC:
19G38 Hermitian \(K\)-theory, relations with \(K\)-theory of rings
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
14F42 Motivic cohomology; motivic homotopy theory
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