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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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##### References:
 [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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