On the nonexistence of elements of Kervaire invariant one

Pages 1-262 from Volume 184 (2016), Issue 1 by M. A. Hill, M. J. Hopkins, D. C. Ravenel

Abstract

We show that the Kervaire invariant one elements $\theta_{j}\in\pi_{2^{j+1}-2}S^{0}$ exist only for $j\le 6$. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions $2$, $6$, $14$, $30$, $62$, and possibly $126$. Except for dimension $126$ this resolves a longstanding problem in algebraic topology.

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      note = {S{é}minaire de g{é}om{é}trie alg{é}brique du Bois Marie 1960--61. [Algebraic Geometry Seminar of Bois Marie 1960-61], with two papers by M. Raynaud, updated and annotated reprint of the 1971 original [{\em Lecture Notes in Math.} {\bf 224},
      Springer, Berlin; \mr{0354651}]},
      editor={A. Grothendieck},
      publisher = {Société Mathématique de France},
      address = {Paris},
      year = {2003},
      pages = {xviii+327},
      isbn = {2-85629-141-4},
      mrclass = {14E20 (14-06 14F35)},
      mrnumber = {2017446},
      zblnumber = {1039.14001},
      }
  • [MR3290092] Go to document M. A. Hill and M. J. Hopkins, "Equivariant multiplicative closure," in Algebraic Topology: Applications and New Directions, Amer. Math. Soc., Providence, RI, 2014, vol. 620, pp. 183-199.
    @incollection{MR3290092, mrkey = {3290092},
      author = {Hill, M. A. and Hopkins, M. J.},
      title = {Equivariant multiplicative closure},
      booktitle = {Algebraic Topology: Applications and New Directions},
      series = {Contemp. Math.},
      volume = {620},
      pages = {183--199},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2014},
      mrclass = {55P43 (55P60 55P91)},
      mrnumber = {3290092},
      mrreviewer = {Steven R. Costenoble},
      doi = {10.1090/conm/620/12372},
      zblnumber = {06520465},
     }
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      author = {Hirschhorn, Philip S.},
      title = {Model Categories and their Localizations},
      series = {Math. Surveys Monogr.},
      number = {99},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {2003},
      pages = {xvi+457},
      isbn = {0-8218-3279-4},
      mrclass = {18G55 (55P60 55U35)},
      mrnumber = {1944041},
      mrreviewer = {David A. Blanc},
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      }
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      author = {Hopkins, Michael J. and Goerss, P.},
      title = {Multiplicative stable homotopy theory},
      note = {unpublished manuscript},
      year = {1996},
      }
  • [HS] Go to document M. J. Hopkins and J. H. Smith, "Nilpotence and stable homotopy theory. II," Ann. of Math., vol. 148, iss. 1, pp. 1-49, 1998.
    @article{HS, mrkey = {1652975},
      author = {Hopkins, Michael J. and Smith, Jeffrey H.},
      title = {Nilpotence and stable homotopy theory. {II}},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {148},
      year = {1998},
      number = {1},
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      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {55P42 (55N20 55Q10)},
      mrnumber = {1652975},
      mrreviewer = {David A. Blanc},
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      }
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      }
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      author = {Hovey, Mark},
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      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {1999},
      pages = {xii+209},
      isbn = {0-8218-1359-5},
      mrclass = {55U35 (18D15 18G30 18G55)},
      mrnumber = {1650134},
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      }
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      author = {Hu, Po and Kriz, Igor},
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      coden = {TPLGAF},
      mrclass = {55T15 (19L47 55N22 55N91 55P42 55P91)},
      mrnumber = {1808224},
      mrreviewer = {J. P. C. Greenlees},
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      author = {Illman, S{ö}ren},
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      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {233},
      year = {1978},
      number = {3},
      pages = {199--220},
      issn = {0025-5831},
      mrclass = {57D05 (57E15)},
      mrnumber = {0500993},
      mrreviewer = {G. E. Bredon},
      doi = {10.1007/BF01405351},
      zblnumber = {0359.57001},
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    @article{MR696520, mrkey = {0696520},
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      title = {The equivariant triangulation theorem for actions of compact {L}ie groups},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {262},
      year = {1983},
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      pages = {487--501},
      issn = {0025-5831},
      coden = {MAANA},
      mrclass = {57S15 (57R05)},
      mrnumber = {0696520},
      mrreviewer = {Karl Heinz Dovermann},
      doi = {10.1007/BF01456063},
      zblnumber = {0488.57014},
      }
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      author = {Johnstone, Peter T.},
      title = {Sketches of an Elephant: A Topos Theory Compendium. {V}ol. 1},
      series = {Oxford Logic Guides},
      volume = {43},
      publisher = {The Clarendon Press, Oxford University Press},
      address = {New York},
      year = {2002},
      pages = {xxii+468+71},
      isbn = {0-19-853425-6},
      mrclass = {18B25 (18-02)},
      mrnumber = {1953060},
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      zblnumber = {1071.18001},
      }
  • [MR508888] Go to document J. D. S. Jones, "The Kervaire invariant of extended power manifolds," Topology, vol. 17, iss. 3, pp. 249-266, 1978.
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      author = {Jones, John D. S.},
      title = {The {K}ervaire invariant of extended power manifolds},
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      volume = {17},
      year = {1978},
      number = {3},
      pages = {249--266},
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      coden = {TPLGAF},
      mrclass = {55Q40},
      mrnumber = {0508888},
      mrreviewer = {Stanley Kochman},
      doi = {10.1016/0040-9383(78)90029-0},
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      }
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      author = {Kelly, Gregory Maxwell},
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      number = {64},
      publisher = {Cambridge Univ. Press},
      address = {Cambridge},
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      isbn = {0-521-28702-2},
      mrclass = {18-02 (18D20)},
      mrnumber = {0651714},
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  • [kervaire60] Go to document M. A. Kervaire, "A manifold which does not admit any differentiable structure," Comment. Math. Helv., vol. 34, pp. 257-270, 1960.
    @article{kervaire60, mrkey = {0139172},
      author = {Kervaire, Michel A.},
      title = {A manifold which does not admit any differentiable structure},
      journal = {Comment. Math. Helv.},
      fjournal = {Commentarii Mathematici Helvetici},
      volume = {34},
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      pages = {257--270},
      issn = {0010-2571},
      mrclass = {57.10},
      mrnumber = {0139172},
      mrreviewer = {R. Bott},
      url = {http://retro.seals.ch/digbib/view2?pid=com-001:1960:34::287},
      zblnumber = {0145.20304},
      }
  • [kervaire63:_group] Go to document M. A. Kervaire and J. W. Milnor, "Groups of homotopy spheres. I," Ann. of Math., vol. 77, pp. 504-537, 1963.
    @article{kervaire63:_group, mrkey = {0148075},
      author = {Kervaire, Michel A. and Milnor, John W.},
      title = {Groups of homotopy spheres. {I}},
      journal = {Ann. of Math.},
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      volume = {77},
      year = {1963},
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      mrclass = {57.10},
      mrnumber = {0148075},
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      doi = {10.2307/1970128},
      zblnumber = {0115.40505},
      }
  • [MR0222890] Go to document P. S. Landweber, "Conjugations on complex manifolds and equivariant homotopy of $MU$," Bull. Amer. Math. Soc., vol. 74, pp. 271-274, 1968.
    @article{MR0222890, mrkey = {0222890},
      author = {Landweber, Peter S.},
      title = {Conjugations on complex manifolds and equivariant homotopy of {$MU$}},
      journal = {Bull. Amer. Math. Soc.},
      fjournal = {Bulletin of the American Mathematical Society},
      volume = {74},
      year = {1968},
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      mrclass = {55.36 (57.00)},
      mrnumber = {0222890},
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      }
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    @article{Land:Hom, mrkey = {0423332},
      author = {Landweber, Peter S.},
      title = {Homological properties of comodules over {$M{\rm U}\sb\ast (M{\rm U})$} and {BP{$\sb\ast $}}({BP})},
      journal = {Amer. J. Math.},
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      mrnumber = {0423332},
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    @incollection{MR0335598, mrkey = {0335598},
      author = {Laplaza, Miguel L.},
      title = {Coherence for distributivity},
      booktitle = {Coherence in Categories},
      pages = {29--65},
      series = {Lecture Notes in Math.},
      volume = {281},
      publisher = {Springer-Verlag},
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      mrnumber = {0335598},
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      doi = {10.1007/BFb0059555},
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    @incollection{MR802786, mrkey = {0802786},
      author = {Levine, J. P.},
      title = {Lectures on groups of homotopy spheres},
      booktitle = {Algebraic and Geometric Topology},
      venue = {{N}ew {B}runswick, {N}.{J}., 1983},
      series = {Lecture Notes in Math.},
      volume = {1126},
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      mrclass = {57R60 (57R65)},
      mrnumber = {0802786},
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      }
  • [MR598689] Go to document G. Lewis, J. P. May, and J. McClure, "Ordinary $RO(G)$-graded cohomology," Bull. Amer. Math. Soc., vol. 4, iss. 2, pp. 208-212, 1981.
    @article{MR598689, mrkey = {0598689},
      author = {Lewis, G. and May, J. P. and McClure, J.},
      title = {Ordinary {$RO(G)$}-graded cohomology},
      journal = {Bull. Amer. Math. Soc.},
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      volume = {4},
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      issn = {0273-0979},
      coden = {BAMOAD},
      mrclass = {55N25 (55P42)},
      mrnumber = {0598689},
      mrreviewer = {Stewart B. Priddy},
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      zblnumber = {0477.55009},
      }
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      author = {Lewis, Jr., L. G. and May, J. P. and Steinberger, M. and McClure, J. E.},
      title = {Equivariant Stable Homotopy Theory},
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      volume = {1213},
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      isbn = {3-540-16820-6},
      mrclass = {55-02 (55Nxx 55Pxx 57S99)},
      mrnumber = {0866482},
      mrreviewer = {T. tom Dieck},
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     }
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      volume = {81},
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      pages = {380--387},
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      mrclass = {14.50 (10.67)},
      mrnumber = {0172878},
      mrreviewer = {P. Roquette},
      doi = {10.2307/1970622},
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      author = {Mac Lane, Saunders and Moerdijk, Ieke},
      title = {Sheaves in Geometry and Logic, A First Introduction toTopos Theory},
      series = {Universitext},
      note = {corrected reprint of the 1992 edition},
      publisher = {Springer-Verlag},
      year = {1994},
      pages = {xii+629},
      isbn = {0-387-97710-4},
      mrclass = {03G30 (18B25 54B40)},
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      author = {Mac Lane, Saunders},
      title = {Categories for the Working Mathematician},
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      volume = {5},
      edition = {Second},
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      isbn = {0-387-98403-8},
      mrclass = {18-02},
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  • [MR0214072] Go to document M. Mahowald and M. Tangora, "Some differentials in the Adams spectral sequence," Topology, vol. 6, pp. 349-369, 1967.
    @article{MR0214072, mrkey = {0214072},
      author = {Mahowald, Mark and Tangora, Martin},
      title = {Some differentials in the {A}dams spectral sequence},
      journal = {Topology},
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      volume = {6},
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      mrclass = {55.52},
      mrnumber = {0214072},
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      doi = {10.1016/0040-9383(67)90023-7},
      zblnumber = {0166.19004},
      }
  • [MR1922205] Go to document M. A. Mandell and J. P. May, "Equivariant orthogonal spectra and $S$-modules," Mem. Amer. Math. Soc., vol. 159, iss. 755, p. x, 2002.
    @article{MR1922205, mrkey = {1922205},
      author = {Mandell, M. A. and May, J. P.},
      title = {Equivariant orthogonal spectra and {$S$}-modules},
      journal = {Mem. Amer. Math. Soc.},
      fjournal = {Memoirs of the American Mathematical Society},
      volume = {159},
      year = {2002},
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      coden = {MAMCAU},
      mrclass = {55P91 (18E30 55P42 55P43 55P48)},
      mrnumber = {1922205},
      mrreviewer = {J. P. C. Greenlees},
      doi = {10.1090/memo/0755},
      zblnumber = {1025.55002},
      }
  • [MR1806878] Go to document M. A. Mandell, J. P. May, S. Schwede, and B. Shipley, "Model categories of diagram spectra," Proc. London Math. Soc., vol. 82, iss. 2, pp. 441-512, 2001.
    @article{MR1806878, mrkey = {1806878},
      author = {Mandell, M. A. and May, J. P. and Schwede, S. and Shipley, B.},
      title = {Model categories of diagram spectra},
      journal = {Proc. London Math. Soc.},
      fjournal = {Proceedings of the London Mathematical Society. Third Series},
      volume = {82},
      year = {2001},
      number = {2},
      pages = {441--512},
      issn = {0024-6115},
      coden = {PLMTAL},
      mrclass = {55P42 (18A25 18E30 55U35)},
      mrnumber = {1806878},
      mrreviewer = {Mark Hovey},
      doi = {10.1112/S0024611501012692},
      zblnumber = {1017.55004},
      }
  • [MR2066508] Go to document M. A. Mandell, "Equivariant symmetric spectra," in Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic $K$-Theory, Providence, RI: Amer. Math. Soc., 2004, vol. 346, pp. 399-452.
    @incollection{MR2066508, mrkey = {2066508},
      author = {Mandell, Michael A.},
      title = {Equivariant symmetric spectra},
      booktitle = {Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic {$K$}-Theory},
      series = {Contemp. Math.},
      volume = {346},
      pages = {399--452},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {2004},
      mrclass = {55P91 (55P42)},
      mrnumber = {2066508},
      mrreviewer = {J. P. C. Greenlees},
      doi = {10.1090/conm/346/06297},
      zblnumber = {1074.55003},
      }
  • [may64:_lie_hopf] Go to document P. J. May, "The cohomology of restricted Lie algebras and of Hopf algebras," Bull. Amer. Math. Soc., vol. 71, pp. 372-377, 1965.
    @article{may64:_lie_hopf, mrkey = {0185595},
      author = {May, J. Peter},
      title = {The cohomology of restricted {L}ie algebras and of {H}opf algebras},
      journal = {Bull. Amer. Math. Soc.},
      fjournal = {Bulletin of the American Mathematical Society},
      volume = {71},
      year = {1965},
      pages = {372--377},
      issn = {0002-9904},
      mrclass = {55.34},
      mrnumber = {0185595},
      mrreviewer = {Bruno Harris},
      doi = {10.1090/S0002-9904-1965-11300-3},
      zblnumber = {0134.19103},
      }
  • [MR1413302] J. P. May, Equivariant Homotopy and Cohomology Theory, Providence, RI: Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, 1996, vol. 91.
    @book{MR1413302, mrkey = {1413302},
      author = {May, J. P.},
      title = {Equivariant Homotopy and Cohomology Theory},
      series = {CBMS Regional Conf. Ser. Math.},
      volume = {91},
      note = {with contributions by M. Cole, G. Comeza{ñ}a, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner},
      publisher = {Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society},
      address = {Providence, RI},
      year = {1996},
      pages = {xiv+366},
      isbn = {0-8218-0319-0},
      mrclass = {55P91 (18G99 55N91 55U35)},
      mrnumber = {1413302},
      mrreviewer = {Ian Hambleton},
      zblnumber = {0890.55001},
      }
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Keywords
, , , ,
Mathematical Subject Classification
Primary: , , , , , ,
DOI
https://doi.org/10.4007/annals.2016.184.1.1
MR
3505179
zbMATH
06605831
Milestones
Received: 20 November 2010
Revised: 22 September 2015
Accepted: 30 September 2015
Published online: 1 July 2016

Authors

M. A. Hill

University of Virginia, Charlottesville, VA

Current address:

University of California, Los Angeles, Los Angeles, CA M. J. Hopkins

Harvard University, Cambridge, MA

D. C. Ravenel

University of Rochester, Rochester, NY