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

  • Michael A. Hill , Michael J. Hopkins and Douglas C. Ravenel EMAIL logo
Published/Copyright: May 27, 2016
Forum Mathematicum
From the journal Forum Mathematicum Volume 29 Issue 2

Abstract

We describe the slice spectral sequence of a 32-periodic C4-spectrum K[2] related to the C4 norm MU((C4))=NC2C4⁢MUℝ of the real cobordism spectrum MUℝ. 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 Kℝ 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 j≥7.

Keywords: Equivariant stable homotopy theory; Kervaire invariant; Mackey functor; slice spectral sequence
MSC 2010: 55Q10; 55Q91; 55P42; 55R45; 55T99

Communicated by Frederick R. Cohen

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

References

[1] Adams J. F., Stable Homotopy and Generalised Homology, Chicago Lectures in Math., University of Chicago Press, Chicago, 1974. Search in 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. 10.1007/BFb0075584Search in Google Scholar

[3] Atiyah M. F., K-theory and reality, Q. J. Math. Ser. (2) 17 (1966), 367–386. 10.1093/qmath/17.1.367Search in Google Scholar

[4] Dugger D., An Atiyah–Hirzebruch spectral sequence for KR-theory, K-Theory 35 (2005), no. 3–4, 213–256. 10.1007/s10977-005-1552-9Search in 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. 10.1016/B978-044481779-2/50009-2Search in 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. Search in Google Scholar

[7] Hill M. A., Hopkins M. J. and Ravenel D. C., The slice spectral sequence for RO⁢(Cpn)-graded suspensions of H⁢𝐙¯ I, preprint 2016. 10.1007/s40590-016-0129-3Search in Google Scholar

[8] Lewis, Jr. L. G., The R⁢O⁢(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. 10.1007/BFb0083034Search in Google Scholar

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

[10] Thévenaz J. and Webb P., The structure of Mackey functors, Trans. Amer. Math. Soc. 347 (1995), no. 6, 1865–1961. 10.1090/S0002-9947-1995-1261590-5Search in Google Scholar

Received: 2016-1-22
Published Online: 2016-5-27
Published in Print: 2017-3-1

© 2017 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. A note on the Hilali conjecture
  3. Finite groups have more conjugacy classes
  4. The gauge action, DG Lie algebras and identities for Bernoulli numbers
  5. Unitary embeddings of finite loop spaces
  6. Topological complexity of planar polygon spaces with small genetic code
  7. Numerical semigroups in a problem about cost-effective transport
  8. On the smallest simultaneous power nonresidue modulo a prime
  9. Decompositions of the higher order polars of plane branches
  10. A formula for the expected volume of the Wiener sausage with constant drift
  11. The slice spectral sequence for the C4 analog of real K-theory
  12. Incidences between points and generalized spheres over finite fields and related problems
  13. Mean values of the Witten L-function for SU(2)
  14. A direct computation of the cohomology of the braces operad
  15. Abelian varieties over finite fields as basic abelian varieties
https://doi.org/10.1515/forum-2016-0017
Keywords for this article
Equivariant stable homotopy theory; Kervaire invariant; Mackey functor; slice spectral sequence

Articles in the same Issue

  1. Frontmatter
  2. A note on the Hilali conjecture
  3. Finite groups have more conjugacy classes
  4. The gauge action, DG Lie algebras and identities for Bernoulli numbers
  5. Unitary embeddings of finite loop spaces
  6. Topological complexity of planar polygon spaces with small genetic code
  7. Numerical semigroups in a problem about cost-effective transport
  8. On the smallest simultaneous power nonresidue modulo a prime
  9. Decompositions of the higher order polars of plane branches
  10. A formula for the expected volume of the Wiener sausage with constant drift
  11. The slice spectral sequence for the C4 analog of real K-theory
  12. Incidences between points and generalized spheres over finite fields and related problems
  13. Mean values of the Witten L-function for SU(2)
  14. A direct computation of the cohomology of the braces operad
  15. Abelian varieties over finite fields as basic abelian varieties
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