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THE TOM DIECK SPLITTING THEOREM IN EQUIVARIANT MOTIVIC HOMOTOPY THEORY

Part of: Homotopy theory (Co)homology theory

Published online by Cambridge University Press:  10 November 2021

David Gepner Open the ORCID record for David Gepner [Opens in a new window]  and
Jeremiah Heller
David Gepner*
Affiliation:
Johns Hopkins University, Department of Mathematics, 404 Krieger Hall, 3400 N. Charles Street, Baltimore, MD 21218, USA
Jeremiah Heller
Affiliation:
University of Illinois at Urbana-Champaign, Department of Mathematics, 1409 W. Green Street, Urbana, IL 61801, USA (jbheller@illinois.edu)
*
dgepner1@jhu.edu
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Abstract

We establish, in the setting of equivariant motivic homotopy theory for a finite group, a version of tom Dieck’s splitting theorem for the fixed points of a suspension spectrum. Along the way we establish structural results and constructions for equivariant motivic homotopy theory of independent interest. This includes geometric fixed-point functors and the motivic Adams isomorphism.

Keywords

motivic homotopy theoryequivariant homotopy theoryAdams isomorphismtom Dieck splitting

MSC classification

Primary: 14F42: Motivic cohomology; motivic homotopy theory 55P91: Equivariant homotopy theory
Secondary: 55P42: Stable homotopy theory, spectra 55P92: Relations between equivariant and nonequivariant homotopy theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 3 , May 2023 , pp. 1181 - 1250
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Adams, J. F., Prerequisites (on equivariant stable homotopy) for Carlsson’s lecture, in Algebraic Topology, Aarhus 1982 (Aarhus, 1982) , Lecture Notes in Mathematics , 1051, pp. 483–532 (Springer,Berlin, 1984).Google Scholar
Antieau, B. and Elmanto, E., A primer for unstable motivic homotopy theory, in Surveys on Recent Developments in Algebraic Geometry, Proceedings of Symposia in Pure Mathematics, 95, pp. 305–370 (American Mathematical Society, Providence, RI, 2017).Google Scholar
Artin, M., Grothendieck, A. and Verdier, J.-L., Theorie de Topos et Cohomologie Etale des Schemas I, II, III , Lecture Notes in Mathematics , 269, 270, 305 (Springer, Springer-Verlag Berlin Heidelberg, 1971).Google Scholar
Asok, A., Hoyois, M. and Wendt, M., Affine representability results in ${A}^1$ -homotopy theory, I: Vector bundles, Duke Math. J. 166(10) (2017), 1923–1953.CrossRefGoogle Scholar
Bachmann, T., Motivic and real étale stable homotopy theory, Compos. Math. 154(5) (2018), 883–917.CrossRefGoogle Scholar
Bachmann, T., Elmanto, E. and Heller, J., Motivic colimits and extended powers, Preprint, 2021, https://arxiv.org/abs/2104.01057.Google Scholar
Bachmann, T. and Hoyois, M., Norms in motivic homotopy theory, Preprint, 2017, arXiv:1711.03061.Google Scholar
Carlsson, G., Equivariant stable homotopy and Segal’s Burnside ring conjecture, Ann. of Math. (2) 120(2) (1984), 189–224.CrossRefGoogle Scholar
Carlsson, G. and Joshua, R., Equivariant motivic homotopy theory, Preprint, 2014, https://arxiv.org/abs/1404.1597.Google Scholar
Conrad, B., Gabber, O. and Prasad, G., Pseudo-Reductive Groups , New Mathematical Monographs , 17 (Cambridge University Press, Cambridge, UK, 2010).Google Scholar
Deligne, P., Voevodsky’s lectures on motivic cohomology 2000/2001, in Algebraic topology, Abel Symposia, 4, pp. 355–409 (Springer, Berlin, 2009).CrossRefGoogle Scholar
Demazure, M. and Grothendieck, A., Schémas en groupes I, II, III , Lecture Notes in Mathematics, 151, 152, 153 (Springer, Springer-Verlag Berlin Heidelberg, 1971).Google Scholar
Dieck, T. Tom, Orbittypen und äquivariante Homologie. II, Arch. Math. (Basel) 26(6) (1975), 650–662.Google Scholar
Fausk, H., Hu, P. and May, J. P., Isomorphisms between left and right adjoints, Theory Appl. Categ. 11(4) (2003), 107–131.Google Scholar
Gepner, D., Groth, M. and Nikolaus, T., Universality of multiplicative infinite loop space machines, Algebr. Geom. Topol. 15(6) (2015), 3107–3153.CrossRefGoogle Scholar
Gepner, D. and Kock, J., Univalence in locally cartesian closed $\infty$ -categories, Forum Math. 29(3) (2017), 617–652.CrossRefGoogle Scholar
Greenlees, J. P. C. and May, J. P., Generalized Tate Cohomology, Memoirs of the American Mathematical Society, 113 (American Mathematical Society, Providence, RI, 1995).Google Scholar
Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique IV , Publications Mathématiques, Institut des Hautes Études Scientifiques, 20, 24, 28, 32, (Publisher, Location), 1964–1967).CrossRefGoogle Scholar
Heller, J., Krishna, A. and Østvær, P. A., Motivic homotopy theory of group scheme actions, J. Topol. 8(4) (2015), 1202–1236.CrossRefGoogle Scholar
Heller, J., Voineagu, M. and Østvær, P. A., Topological comparison theorems for Bredon motivic cohomology, Trans. Amer. Math. Soc. 371(4) (2019), 2875–2921.Google Scholar
Herrmann, P., Equivariant motivic homotopy theory, Preprint, 2013, arXiv:1312.0241.Google Scholar
Hill, M. A., Hopkins, M. J. and Ravenel, D. C., On the nonexistence of elements of Kervaire invariant one, Ann. of Math. (2) 184(1) (2016), 1–262.Google Scholar
Hoyois, M., Cdh descent in equivariant homotopy K-theory, Preprint, 2016, arXiv:1604.06410.Google Scholar
Hoyois, M., The six operations in equivariant motivic homotopy theory, Adv. Math. 305 (2017), 197–279.Google Scholar
Hu, P., Kriz, I. and Ormsby, K., The homotopy limit problem for Hermitian K-theory, equivariant motivic homotopy theory and motivic Real cobordism, Adv. Math. 228(1) (2011), 434–480.Google Scholar
Khan, A., The Morel-Voevodsky localization theorem in spectral algebraic geometry, Preprint, 2016, arXiv:1610.06871.Google Scholar
Lewis, L. G. Jr., May, J. P., Steinberger, M. and McClure, J. E., Equivariant Stable Homotopy Theory , Lecture Notes in Mathematics, 1213 (Springer, Berlin, 1986). With contributions by J. E. McClure.Google Scholar
Lurie, J., Higher Topos Theory , Annals of Mathematics Studies, 170 (Princeton University Press, Princeton, NJ, 2009).Google Scholar
Lurie, J., Higher algebra. 2012. Version dated 18 September 2017. https://www. math.ias.edu/~lurie/papers/HA.pdf.Google Scholar
Morel, F., ${A}^1$ -algebraic topology, in International Congress of Mathematicians, Vol. II, pp. 1035–1059 (European Mathematical Society, Zürich, 2006).Google Scholar
Morel, F., ${A}^1$ -Algebraic Topology over a Field, Lecture Notes in Mathematics, 2052 (Springer, Heidelberg, 2012).Google Scholar
Morel, F. and Voevodsky, V., ${A}^1$ -homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci. 90 (1999), 45–143.CrossRefGoogle Scholar
Reich, H. and Varisco, M., On the Adams isomorphism for equivariant orthogonal spectra, Algebr. Geom. Topol. 16(3) (2016), 1493–1566.CrossRefGoogle Scholar
Robalo, M., $K$ -theory and the bridge from motives to noncommutative motives, Adv. Math. 269 (2015), 399–550.CrossRefGoogle Scholar
Rydh, D., Existence and properties of geometric quotients, J. Algebraic Geom. 22(4) (2013), 629–669.CrossRefGoogle Scholar
Segal, G. B., Equivariant stable homotopy theory, in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 59–63 (Gauthier-Villars, Paris, 1971).Google Scholar
The Stacks Project Authors, ‘Stacks project’, 2018, https://stacks.math. columbia.edu.Google Scholar
Thomason, R. W., Algebraic $K$ -theory of group scheme actions, in Algebraic Topology and Algebraic $K$ -Theory (Princeton, N.J., 1983), Annals of Mathematics Studies, 113, pp. 539–563 (Princeton University Press, Princeton, NJ, 1987).Google Scholar
Totaro, B., The Chow ring of a classifying space, in Algebraic $K$ -Theory (Seattle, WA, 1997), Proceedings of Symposia in Pure Mathematics, 67, pp. 249–281 (American Mathematical Society, Providence, RI, 1999).Google Scholar
Voevodsky, V., Homotopy theory of simplicial sheaves in completely decomposable topologies, J. Pure Appl. Algebra 214(8) (2010), 1384–1398.CrossRefGoogle Scholar