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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.
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
Full Text: DOI
[1] Cazanave, C., Algebraic homotopy classes of rational functions, Ann. Sci. Éc. Norm. Supér., Sér. 4, 45, 4, 511-534 (2012) · Zbl 1419.14025
[2] Conner, P. E.; Floyd, E. E., The Relation of Cobordism to K-Theories (1966), Berlin: Springer, Berlin · Zbl 0161.42802
[3] Morel, F., \( \bar \varphi :{{\mathop{\rm MSp}\nolimits}_*}^{[*]}(X,U){ \otimes_{{\rm{MSp}}_0^{[2*]}({\rm{pt}})}}{\rm{KO}}_0^{[2*]}({\rm{pt}}) \cong{\rm{K}}{{\rm{O}}_*}^{[*]}(X,U)\)-Algebraic Topology over a Field (2012), Berlin: Springer, Berlin
[4] Nenashev, A., Gysin maps in Balmer-Witt theory, J. Pure Appl. Algebra, 211, 1, 203-221 (2007) · Zbl 1140.11024
[5] Panin, I., Oriented cohomology theories of algebraic varieties. II (after I. Panin and A. Smirnov), Homology, Homotopy Appl., 11, 1, 349-405 (2009) · Zbl 1169.14016
[6] Panin, I.; Pimenov, K.; Röndigs, O., On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory, Invent. Math., 175, 2, 435-451 (2009) · Zbl 1205.14023
[7] I. Panin and C. Walter, “On the algebraic cobordism spectra MSL and MSp,” arXiv: 1011.0651 [math.AG].
[8] Panin, I.; Walter, C., On the motivic commutative ring spectrum BO, St. Petersbg. Math. J., 30, 6, 933-972 (2019) · Zbl 1428.14011
[9] I. Panin and C. Walter, “Quaternionic Grassmannians and Borel classes in algebraic geometry,” arXiv: 1011.0649 [math.AG].
[10] Schlichting, M., Hermitian K-theory of exact categories, J. K-Theory, 5, 1, 105-165 (2010) · Zbl 1328.19009
[11] Schlichting, M., The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes, Invent. Math., 179, 2, 349-433 (2010) · Zbl 1193.19005
[12] Schlichting, M., Hermitian K-theory, derived equivalences and Karoubi’s fundamental theorem, J. Pure Appl. Algebra, 221, 7, 1729-1844 (2017) · Zbl 1360.19008
[13] Voevodsky, V., A^1-homotopy theory, Doc. Math., Extra Vol. ICM Berlin 1998, I, 579-604 (1998) · Zbl 0907.19002
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