On Completion and the Evenness Conjecture for Homotopical Equivariant Cobordism

Abstract

We give counterexamples to the evenness conjecture for homotopical equivariant cobordism. To this end, we prove a completion theorem for certain complex cobordism modules which does not involve higher derived functors. A key step in the proof is provided by a certain new relation between Mackey and Borel cohomology.

Keywords

Complex cobordism / Equivariant cobordism / Equivariant spectra / Completion theorem / Evenness conjecture

References

[1] ÁngelA, SampertonE, SegoviaC, UribeB. Oriented and unitary equivariant bordism of surfaces. Alg. Geom. Topol., 2024, 2431623-1654.

[2] AtiyahMF, SegalGB. Equivariant K-theory and completion. J. Diff. Geom., 1969, 3: 1-18

[3] Beilinson, A. A., Bernstein, J., Deligne, P.: Analyse et topologie sur les espaces singuliers, I. Astérisque 100, 172 pp. (1982)

[4] Blumberg, A., Mandell, M.: K\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K$$\end{document}-theoretic Tate–Poitou duality and the fiber of the cyclotomic trace. Invent. Math. 221(2), 397–419 (2020)

[5] Boardman, J.M.: Conditionally convergent spectral sequences. In: Homotopy Invariant Algebraic Structures (Baltimore, MD, 1998), Contemp. Math., vol. 239, pp. 49–84. American Mathematical Society, Providence, RI (1999)

[6] Bogomolov, F.A.: The Brauer group of quotient spaces of linear representations. Math. USSR Izv. 30(3), 455–485 (1988)

[7] BröckerT, HookEC. Stable equivariant bordism. Math. Z., 1972, 129: 269-277.

[8] Chen, Y., Ma, R.: Bogomolov multipliers of some groups of order p6\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^6$$\end{document}. Commun. Algebra 49(1), 242–255 (2021)

[9] Dress, A.W.M.: Contributions to the theory of induced representations. In: Algebraic K-Theory II, “Classical” Algebraic K-Theory, and Connections with Arithmetic (Seattle, WA, USA 1972), Lecture Notes in Math., vol. 342, pp. 181–240. Springer, Berlin (1973)

[10] Dugger, D.: Multiplicative structures on homotopy spectral sequences II. arXiv:math/0305187 (2003)

[11] Elmendorf, A.D.: Systems of fixed point sets. Trans. Am. Math. Soc. 277(1), 275–284 (1983)

[12] GreenlessJPC. Some remarks on projective Mackey functors. J. Pure. Appl. Algebra, 1992, 81: 17-38.

[13] Greenlees, J.P.C., May, J.P.: Generalized Tate cohomology. Mem. Am. Math. Soc. 113(543), vii+178 pp. (1995)

[14] GreenleesJPC, MayJP. Localization and completion theorems for MU\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M{\rm U}$$\end{document}-module spectra. Ann. Math. (2), 1997, 1463509-544.

[15] HuP. The equivariant Lazard ring of primary cyclic groups. Trans. Amer. Math. Soc., 2025, 37842881-2921

[16] Hu, P., Kriz, I.: Real-oriented homotopy theory and an analogue of the Adams–Novikov spectral sequence. Topology 40(2), 317–399 (2001)

[17] James, R.: The groups of order p6\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^6$$\end{document} (p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p$$\end{document} an odd prime). Math. Comput. 34(150), 613–637 (1980)

[18] Kriz, I.: Morava K\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K$$\end{document}-theory of classifying spaces: some calculations. Topology 36(6), 1247–1273 (1997)

[19] Kriz, I., Lee, K.P.: Odd-degree elements in the Morava K(n)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K(n)$$\end{document} cohomology of finite groups. Topol. Appl. 103(3), 229–241 (2000)

[20] Kriz, S.: Some remarks on Mackey functors. arXiv:2205.12192 (2022)

[21] La VecchiaM. The completion and local cohomology theorems for complex cobordism for all compact Lie groups. Geom. Topol., 2024, 282627-639

[22] Lewis, L.G.Jr., May, J.P., Steinberger, M., McClure, J.E.: Equivariant Stable Homotopy Theory. Lecture Notes in Math., vol. 1213. Springer, Berlin (1986)

[23] LöfflerP. Bordismengruppen unitärer Torusmannigfaltigkeiten. Manuscr. Math., 1974, 12: 307-327.

[24] May, J. P.: Equivariant Homotopy and Cohomology Theory. CBMS Regional Conf. Ser. in Math., vol. 91. American Mathematical Society, Providence, RI (1996)

[25] Milnor, J.W.: On the cobordism ring Ω*\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Omega ^\ast $$\end{document} and a complex analogue, I. Am. J. Math. 82, 505–521 (1960)

[26] NoetherE. Gleichungen mit vorgeschriebener Gruppe. Math. Ann., 1917, 78: 221-229.

[27] NovikovSP. Some problems in the topology of manifolds connected with the theory of Thom spaces. Soviet Math. Dokl., 1960, 1: 717-720

[28] NovikovSP. Homotopy properties of Thom complexes. Mat. Sb. (N.S.), 1962, 57: 407-442

[29] Ravenel, D.: Morava K-theories and finite groups. In: Symposium on Algebraic Topology in honor of José Adem (Oaxtepec, 1981), Contemp. Math., vol. 12, pp. 289–292. American Mathematical Society, Providence, RI (1982)

[30] Rowlett, R.J.: Bordism of metacyclic group actions. Michigan Math. J. 27(2), 223–233 (1980)

[31] SampertonE. Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds. C. R. Math. Acad. Sci. Paris, 2022, 360: 161-167.

[32] StricklandNP. Complex cobordism of involutions. Geom. Top., 2001, 5: 335-345.

[33] tom DieckT. Bordism of G\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G$$\end{document}-manifolds and integrality theorems. Topology, 1970, 94345-358.

[34] tom Dieck, T.: Kobordismentheorie klassifizierender Räume und Transformationsgruppen. Math. Z. 126, 31–39 (1972)

[35] Uribe, B.: The evenness conjecture in equivariant unitary bordism. In: Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018, Vol. II, Invited Lectures, pp. 1217–1239 World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2018)

Funding

Division of Mathematical Sciences(2023350430)

RIGHTS & PERMISSIONS

Peking University

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.