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Panin, I.; Walter, C.

On the algebraic cobordism spectra \(\mathbf{MSL}\) and \(\mathbf{MSp} \). (English) Zbl 07637768

St. Petersbg. Math. J. 34, No. 1, 109-141 (2023) and Algebra Anal. 34, No. 1, 144-187 (2022).
Summary: The algebraic cobordism spectra \(\mathbf{MSL}\) and \(\mathbf{MSp}\) are constructed. They are commutative monoids in the category of symmetric \(T^{\wedge 2} \)-spectra. The spectrum \(\mathbf{MSp}\) comes with a natural symplectic orientation given either by a tautological Thom class \(th^{\mathbf{MSp}} \in \mathbf{MSp}^{4,2}(\mathbf{MSp}_2)\), or a tautological Borel class \(b_1^{\mathbf{MSp}} \in \mathbf{MSp}^{4,2}(HP^{\infty })\), or any of six other equivalent structures. For a commutative monoid \(E\) in the category \({SH}(S)\), it is proved that the assignment \(\varphi \mapsto \varphi (th^{\mathbf{MSp}})\) identifies the set of homomorphisms of monoids \(\varphi \colon \mathbf{MSp}\to E\) in the motivic stable homotopy category \(SH(S)\) with the set of tautological Thom elements of symplectic orientations of \(E\). A weaker universality result is obtained for \(\mathbf{MSL}\) and special linear orientations. The universality of \(\mathbf{MSp}\) has been used by the authors to prove a Conner-Floyed type theorem. The weak universality of \(\mathbf{MSL}\) has been used by A. Ananyevskiy to prove another version of the Conner-Floyed type theorem.

MSC:

14F42 Motivic cohomology; motivic homotopy theory

Keywords:

\(\mathbf{A}^1\)-homotopy theory; Thom classes; universality theorems
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\textit{I. Panin} and \textit{C. Walter}, St. Petersbg. Math. J. 34, No. 1, 109--141 (2023; Zbl 07637768)
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References:

[1] A. Ananyevskiy, Relationship between algebraic MSL-cobordisms and derived Witt groups, Dokl. Akad. Nauk 448 (2013), no. 5, 503-505; English transl., Dokl. Math. 87 (2013), no. 1, 76-78. 3088614 · Zbl 1291.55008
[2] A. Ananyevskiy, On the relation of special linear algebraic cobordism to Witt groups, Homology Homotopy Appl. 18 (2016), no. 1, 204-230. 3491850 · Zbl 1352.14012
[3] P. Balmer and B. Calm\`es, Witt groups of Grassmann varieties, Algebr. Geom. 21 (2012), no. 4, 601-642. 2957690 · Zbl 1273.14098
[4] B. I. Dundas, O. R\"ondigs, and P. A. stvr, Motivic functors, Doc. Math. 8 (2003), 489-525. 2029171 · Zbl 1042.55006
[5] M. Hovey, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165 (2001), no. 1, 63-127. 1860878 · Zbl 1008.55006
[6] D. C. Isaksen, Flasque model structures for simplicial presheaves, K-Theory 36 (2005), no. 3-4, 371-395. 2275013 · Zbl 1116.18008
[7] J. F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445-552. 1787949 · Zbl 0969.19004
[8] M. Levine and F. Morel, Algebraic cobordism, Springer Monogr. Math., Springer, Berlin, 2007. 2286826 · Zbl 1188.14015
[9] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Math. Monogr., Oxford Univ. Press, New York, 1995. 1354144
[10] F. Morel and V. Voevodsky, \( \mathbf A^1\)-homotopy theory of schemes, Inst. Hautes \'Etudes Sci. Publ. Math. 90 (1999), 45-143. 1813224 · Zbl 0983.14007
[11] I. Panin, Oriented cohomology theories of algebraic varieties, Special issue in honor of Hyman Bass on his seventieth birthday. Pt. III, K-Theory 30 (2003), no. 3, 265-314. 2064242 · Zbl 1047.19001
[12] I. Panin, K. Pimenov, and O. R\"ondigs, A universality theorem for Voevodsky’s algebraic cobordism spectrum, Homology Homotopy Appl. 10 (2008), no. 2, 211-226. 2475610 · Zbl 1162.14013
[13] I. Panin, K. Pimenov, and O. R\"ondigs, On Voevodsky’s algebraic \(K\)-theory spectrum, Algebraic topology, Abel Symp., vol. 4, Springer, Berlin, 2009, pp. 279-330. 2597741 · Zbl 1179.14022
[14] I. Panin and C. Walter, On the motivic commutative ring spectrum \(\mathbf BO,\) Algebra i Analiz 30 (2018), no. 6, 43-96; English transl., St. Petersburg Math. J. 30 (2019), no. 6, 933-972. 3882540 · Zbl 1428.14011
[15] I. Panin and C. Walter, On the relation of symplectic algebraic cobordism to hermitian K-theory, Tr. Mat. Inst. Steklova 307 (2019), 180-192; English transl., Proc. Steklov Inst. Math. 307 (2019), 162-173. 4070064 · Zbl 1442.19012
[16] I. Panin and C. Walter, Quaternionic Grassmannians and Borel classes in algebraic geometry, Algebra i Analiz 33 (2021), no. 1, 136-193. 4219508 · Zbl 1486.14067
[17] G. Vezzosi, Brown-Peterson spectra in stable \(A^1\)-homotopy theory, Rend. Sem. Mat. Univ. Padova 106 (2001), 47-64. 1876212 · Zbl 1165.14308
[18] V. Voevodsky, \( \mathbf A\sp 1\)-homotopy theory, Doc. Math. 1998, Extra Vol. I, pp. 579-604. 1648048 · Zbl 0907.19002
[19] V. Voevodsky, O. R\"ondigs, and P. A. stvr, Voevodsky’s Nordfjordeid lectures\textup : motivic homotopy theory, Motivic homotopy theory, Universitext, Springer, Berlin, 2007, pp. 147-221. 2334215
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