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  • MARC LEVINE (a1)


This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from $\operatorname{GL}_{n}$ or $\operatorname{SL}_{n}$ to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy’s splitting principle reduces questions about characteristic classes of vector bundles in $\operatorname{SL}$ -oriented, $\unicode[STIX]{x1D702}$ -invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\operatorname{SL}_{2}$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.



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Marc Levine would like to thank the DFG for its support through the SFB Transregio 45 and the SPP 1786 “Homotopy theory and algebraic geometry”.



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[1] Ananyevskiy, A., SL-oriented cohomology theories, preprint, 2019, arXiv:1901.01597 [math.AG], [math.AT], [math.KT].
[2] Ananyevskiy, A., On the push-forwards for motivic cohomology theories with invertible stable Hopf element , Manuscripta Math. 150(1–2) (2016), 2144.
[3] Ananyevskiy, A., The special linear version of the projective bundle theorem , Compos. Math. 151(3) (2015), 461501.
[4] Asok, A. and Fasel, J., Comparing Euler classes , Q. J. Math. 67(4) (2016), 603635.
[5] Ayoub, J., Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I , Astérisque 314 (2008), x+466.
[6] Bachmann, T. and Fasel, J., On the effectivity of spectra representing motivic cohomology theories, preprint, 2017, arXiv:1710.00594 math.KT.
[7] Barge, J. and Morel, F., Groupe de Chow des cycles orientés et classe d’Euler des fibrés vectoriels , C. R. Acad. Sci. Paris Sér. I Math. 330(4) (2000), 287290.
[8] Bayer-Fluckiger, E. and Suarez, I., Ideal lattices over totally real number fields and Euclidean minima , Arch. Math. (Basel) 86(3) (2006), 217225.
[9] Becker, J. C. and Gottlieb, D. H., The transfer map and fiber bundles , Topology 14 (1975), 112.
[10] Calmès, B. and Fasel, J., The category of finite Chow–Witt correspondences, preprint, 2017, arXiv:1412.2989, 2014.
[11] Cisinski, D.-C. and Déglise, F., Triangulated categories of mixed motives, preprint, 2012 (version 3), arXiv:0912.2110v3 [math.AG].
[12] Déglise, F. and Fasel, J., MW-motivic complexes, preprint, 2016, arXiv:1708.06095.
[13] Déglise, F. and Fasel, J., The Milnor–Witt motivic ring spectrum and its associated theories, preprint, 2017, arXiv:1708.06102.
[14] Druzhinin, A., Effective Grothendieck–Witt motives of smooth varieties, preprint, 2017, arXiv:1709.06273.
[15] Fasel, J., Groupes de Chow–Witt , Mém. Soc. Math. Fr. (N.S.) 113 (2008), viii + 197.
[16] Fasel, J., The projective bundle theorem for I j -cohomology , J. K-Theory 11(2) (2013), 413464.
[17] Fasel, J. and Srinivas, V., Chow–Witt groups and Grothendieck–Witt groups of regular schemes , Adv. Math. 221 (2009), 302329.
[18] Hoyois, M., A quadratic refinement of the Grothendieck–Lefschetz–Verdier trace formula , Algebr. Geom. Topol. 14 (2014), 36033658.
[19] Hoyois, M., The six operations in equivariant motivic homotopy theory , Adv. Math. 305 (2017), 197279.
[20] Jardine, J. F., Motivic symmetric spectra , Doc. Math. 5 (2000), 445552.
[21] Karoubi, M. and Villamayor, O., K-théorie hermitienne , C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A1237A1240.
[22] Kass, J. L. and Wickelgren, K., The class of Eisenbud–Khimshiashvili–Levine is the local A1 -Brouwer degree , Duke Math. J. 168(3) (2019), 429469.
[23] Kass, J. L. and Wickelgren, K., An arithmetic count of the lines on a smooth cubic surface, preprint, 2017, arXiv:1708.01175 [math.AG].
[24] Levine, M., The intrinsic stable normal cone, preprint, 2017, arXiv:1703.03056 [math.AG].
[25] Levine, M., Toward an enumerative geometry with quadratic forms, preprint, 2017, arXiv:1703.03049 [math.AG].
[26] Levine, M. and Raksit, A., Motivic Gauß-Bonnet formulas, preprint, 2018, arXiv:2374347 [math.AG].
[27] May, J. P., The additivity of traces in triangulated categories , Adv. Math. 163(1) (2001), 3473.
[28] May, J. P., Picard groups, Grothendieck rings, and Burnside rings of categories , Adv. Math. 163(1) (2001), 116.
[29] Morel, F., A1 -algebraic Topology over a Field, Lecture Series Mathematics, 2052 , Springer, Heidelberg, 2012.
[30] Morel, F., Introduction to A1 -homotopy Theory, Lectures Series School on Algebraic K-Theory and its Applications, ICTP, Trieste, 2002.
[31] Morel, F., On the Motivic 𝜋0 of the Sphere Spectrum, Axiomatic, Enriched and Motivic Homotopy Theory, 219–260, NATO Sci. Ser. II Math. Phys. Chem. 131 , Kluwer Acad. Publ., Dordrecht, 2004.
[32] Morel, F. and Voevodsky, V., A1 -homotopy theory of schemes , Publ. Math. Inst. Hautes Études Sci. 90 (1999), 45143.
[33] Okonek, C. and Teleman, A., Intrinsic signs and lower bounds in real algebraic geometry , J. Reine Angew. Math. 688 (2014), 219241.
[34] Panin, I. and Walter, C., Quaternionic Grassmannians and Pontryagin classes in algebraic geometry, preprint, 2010, arXiv:1011.0649.
[35] Scharlau, W., Quadratic and Hermitian Forms, Grundlehren der Mathematischen Wissenschaften, 270, Springer, Berlin, 1985.
[36] Schlichting, M., Hermitian K-Theory of exact catgories , J. K-Theory 5 (2010), 105165.
[37] Serre, J.-P., L’invariant de Witt de la forme Tr(x 2 ) , Comment. Math. Helv. 59(4) (1984), 651676.
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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
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