Geodesic
On the algebraic cobordism spectra $\mathbf{MSL}$ and $\mathbf{MSp}$
Algebra i analiz, Tome 34 (2022) no. 1, pp. 144-187

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\mathrm{th}^{\mathbf{MSp}} \in \mathbf{MSp}^{4,2}(\mathbf{MSp}_2)$, 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 $\varphi \mapsto \varphi(\mathrm{th}^{\mathbf{MSp}})$ 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.
Keywords: $\mathbf{Aff}^{1}$-homotopy theory, Thom classes, universality theorems. 
@article{AA_2022_34_1_a5,
     author = {I. Panin and C. Walter},
     title = {On the algebraic cobordism spectra $\mathbf{MSL}$ and $\mathbf{MSp}$},
     journal = {Algebra i analiz},
     pages = {144--187},
     publisher = {mathdoc},
     volume = {34},
     number = {1},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_2022_34_1_a5/}
}
TY  - JOUR
AU  - I. Panin
AU  - C. Walter
TI  - On the algebraic cobordism spectra $\mathbf{MSL}$ and $\mathbf{MSp}$
JO  - Algebra i analiz
PY  - 2022
SP  - 144
EP  - 187
VL  - 34
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2022_34_1_a5/
LA  - en
ID  - AA_2022_34_1_a5
ER  - 
%0 Journal Article
%A I. Panin
%A C. Walter
%T On the algebraic cobordism spectra $\mathbf{MSL}$ and $\mathbf{MSp}$
%J Algebra i analiz
%D 2022
%P 144-187
%V 34
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2022_34_1_a5/
%G en
%F AA_2022_34_1_a5
I. Panin; C. Walter. On the algebraic cobordism spectra $\mathbf{MSL}$ and $\mathbf{MSp}$. Algebra i analiz, Tome 34 (2022) no. 1, pp. 144-187. http://geodesic.mathdoc.fr/item/AA_2022_34_1_a5/

[1] Ananevskii A., “O sootnoshenii algebraicheskikh MSL-kobordizmov i proizvodnykh grupp Vitta”, Dokl. RAN, 448:5 (2013), 503–505

[2] Ananyevskiy A., “On the relation of special linear algebraic cobordism to Witt groups”, Homology Homotopy Appl., 18:1 (2016), 204–230 | DOI | MR | DOI | MR

[3] Balmer P., Calmès B., “Witt groups of Grassmann varieties”, J. Algebr. Geom., 21:4 (2021), 601–642 | DOI | MR | DOI | MR

[4] Dundas B. I., Röndigs O., Østvær P. A., “Motivic functors”, Doc. Math., 8 (2003), 489–525 | MR | Zbl | MR | Zbl

[5] Hovey M., “Spectra and symmetric spectra in general model categories”, J. Pure Appl. Algebra, 165:1 (2001), 63–127 | DOI | MR | Zbl | DOI | MR | Zbl

[6] Isaksen D. C., “Flasque model structures for simplicial presheaves”, K-Theory, 36:3-4 (2005), 371–395 | DOI | MR | Zbl | DOI | MR | Zbl

[7] Jardine J. F., “Motivic symmetric spectra”, Doc. Math., 5 (2000), 445–552 | MR | MR

[8] Levine M., Morel F., Algebraic cobordism, Springer Monogr. Math., Springer, Berlin, 2007 | Zbl | Zbl

[9] Macdonald I. G., Symmetric functions and Hall polynomials, Oxford Math. Monogr., 2nd ed., Oxford Univ. Press, New York, 1995 | Zbl | Zbl

[10] Morel F., Voevodsky V., “$A^1$-homotopy theory of schemes”, Inst. Hautes Études Sci. Publ. Math., 1999, 45–143 | DOI | Zbl | DOI | Zbl

[11] Panin I., “Oriented cohomology theories of algebraic varieties”, K-Theory, 30:3, Special issue in honor of Hyman Bass on his seventieth birthday. Pt. III (2003), 265–314 | DOI | MR | Zbl | DOI | MR | Zbl

[12] Panin I., Pimenov K., Röndigs O., “A universality theorem for Voevodsky's algebraic cobordism spectrum”, Homology Homotopy Appl., 10:2 (2008), 211–226 | DOI | MR | Zbl | DOI | MR | Zbl

[13] Panin I., Pimenov K., Röndigs O., “On Voevodsky's algebraic $K$-theory spectrum”, Algebraic topology, Abel Symp., v. 4, Springer, Berlin, 2009, 279–330 | DOI | MR | Zbl | DOI | MR | Zbl

[14] Panin I., Walter C., “On the motivic commutative ring spectrum $\mathbf{BO}$, On the motivic commutative ring spectrum BO”, Algebra i analiz, 30:6 (2018), 43–96

[15] Panin I. A., Valter Ch., “O svyazi simplekticheskikh algebraicheskikh kobordizmov i ermitovoi K-teorii”, Tr. Mat. in-ta RAN, 307, 2019, 180–192 | Zbl | Zbl

[16] Panin I., Walter C., “Quaternionic Grassmannians and Borel classes in algebraic geometry”, Algebra i analiz, 33:1 (2021), 136–193 | MR | MR

[17] Vezzosi G., “Brown–Peterson spectra in stable $\mathbb{A}^1$-homotopy theory”, Rend. Sem. Mat. Univ. Padova, 106 (2001), 47–64 | MR | Zbl | MR | Zbl

[18] Voevodsky V., “$\mathbf{A}^1$-homotopy theory”, Doc. Math., v. I, 1998, 579–604 | Zbl | Zbl

[19] Voevodsky V., Röndigs O., Østvær P. A., “Voevodsky's Nordfjordeid lectures: motivic homotopy theory”, Motivic homotopy theory, Universitext, Springer, Berlin, 2007, 147–221 | DOI | MR | DOI | MR