Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

Online
ISSN
1435-5337
See all formats and pricing
More options …
Volume 29, Issue 2

Issues

The slice spectral sequence for the C4 analog of real K-theory

Michael A. Hill / Michael J. Hopkins / Douglas C. Ravenel
Published Online: 2016-05-27 | DOI: https://doi.org/10.1515/forum-2016-0017

Abstract

We describe the slice spectral sequence of a 32-periodic C4-spectrum K[2] related to the C4 norm MU((C4))=NC2C4MU of the real cobordism spectrum MU. We will give it as a spectral sequence of Mackey functors converging to the graded Mackey functor π¯*K[2], complete with differentials and exotic extensions in the Mackey functor structure. The slice spectral sequence for the 8-periodic real K-theory spectrum K was first analyzed by Dugger. The C8 analog of K[2] is 256-periodic and detects the Kervaire invariant classes θj. A partial analysis of its slice spectral sequence led to the solution to the Kervaire invariant problem, namely the theorem that θj does not exist for j7.

Keywords: Equivariant stable homotopy theory; Kervaire invariant; Mackey functor; slice spectral sequence

MSC 2010: 55Q10; 55Q91; 55P42; 55R45; 55T99

References

  • [1]

    Adams J. F., Stable Homotopy and Generalised Homology, Chicago Lectures in Math., University of Chicago Press, Chicago, 1974. Google Scholar

  • [2]

    Adams J. F., Prerequisites (on equivariant stable homotopy) for Carlsson’s lecture, Algebraic Topology (Aarhus 1982), Lecture Notes in Math. 1051, Springer, Berlin (1984), 483–532. Google Scholar

  • [3]

    Atiyah M. F., K-theory and reality, Q. J. Math. Ser. (2) 17 (1966), 367–386. Google Scholar

  • [4]

    Dugger D., An Atiyah–Hirzebruch spectral sequence for KR-theory, K-Theory 35 (2005), no. 3–4, 213–256. Google Scholar

  • [5]

    Greenlees J. P. C. and May J. P., Equivariant stable homotopy theory, Handbook of Algebraic Topology, North-Holland, Amsterdam (1995), 277–323. Google Scholar

  • [6]

    Hill M. A., Hopkins M. J. and Ravenel D. C., The non-existence of elements of Kervaire invariant one, preprint 2010, http://arxiv.org/abs/0908.3724v2.

  • [7]

    Hill M. A., Hopkins M. J. and Ravenel D. C., The slice spectral sequence for RO(Cpn)-graded suspensions of H𝐙¯ I, preprint 2016. Google Scholar

  • [8]

    Lewis, Jr. L. G., The RO(G)-graded equivariant ordinary cohomology of complex projective spaces with linear 𝐙/p actions, Algebraic Topology and Transformation Groups (Göttingen 1987), Lecture Notes in Math. 1361, Springer, Berlin (1988), 53–122. Google Scholar

  • [9]

    Ravenel D. C., Complex Cobordism and Stable Homotopy Groups of Spheres, Pure App. Math. 121, Academic Press, Orlando, 1986. Google Scholar

  • [10]

    Thévenaz J. and Webb P., The structure of Mackey functors, Trans. Amer. Math. Soc. 347 (1995), no. 6, 1865–1961. Google Scholar

About the article


Received: 2016-01-22

Published Online: 2016-05-27

Published in Print: 2017-03-01


The authors were supported by DARPA Grant FA9550-07-1-0555 and NSF Grants DMS-0905160, DMS-1307896.


Citation Information: Forum Mathematicum, Volume 29, Issue 2, Pages 383–447, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2016-0017.

Export Citation

© 2017 by De Gruyter.Get Permission

Comments (0)

Please log in or register to comment.
Log in