The triviality of the 61-stem in the stable homotopy groups of spheres

Abstract

We prove that the 2-primary $\pi_{61}$ is zero. As a consequence, the Kervaire invariant element $\theta_5$ is contained in the strictly defined 4-fold Toda bracket $\langle 2, \theta_4, \theta_4, 2\rangle$.
Our result has a geometric corollary: the 61-sphere has a unique smooth structure, and it is the last odd dimensional case — the only ones are $S^1, S^3, S^5$ and $S^{61}$. Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential $d_3(D_3) = B_3$. We prove this differential by introducing a new technique based on the algebraic and geometric Kahn-Priddy theorems. The success of this technique suggests a theoretical way to prove Adams differentials in the sphere spectrum inductively by use of differentials in truncated projective spectra.

  • [Ada1] Go to document J. F. Adams, "On the structure and applications of the Steenrod algebra," Comment. Math. Helv., vol. 32, pp. 180-214, 1958.
    @ARTICLE{Ada1,
      author = {Adams, J. F.},
      title = {On the structure and applications of the {S}teenrod algebra},
      journal = {Comment. Math. Helv.},
      fjournal = {Commentarii Mathematici Helvetici},
      volume = {32},
      year = {1958},
      pages = {180--214},
      issn = {0010-2571},
      mrclass = {55.00},
      mrnumber = {0096219},
      mrreviewer = {F. P. Peterson},
      doi = {10.1007/BF02564578},
      url = {http://dx.doi.org/10.1007/BF02564578},
      zblnumber = {0083.17802},
      }
  • [Ada2] Go to document J. F. Adams, "On the non-existence of elements of Hopf invariant one," Ann. of Math. (2), vol. 72, pp. 20-104, 1960.
    @ARTICLE{Ada2,
      author = {Adams, J. F.},
      title = {On the non-existence of elements of {H}opf invariant one},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {72},
      year = {1960},
      pages = {20--104},
      issn = {0003-486X},
      mrclass = {55.40},
      mrnumber = {0141119},
      mrreviewer = {M. A. Kervaire},
      doi = {10.2307/1970147},
      url = {http://dx.doi.org/10.2307/1970147},
      zblnumber = {0096.17404},
      }
  • [Aub] Go to document M. Aubry, "Calculs de groupes d’homotopie stables de la sphère, par la suite spectrale d’Adams-Novikov," Math. Z., vol. 185, iss. 1, pp. 45-91, 1984.
    @ARTICLE{Aub,
      author = {Aubry, Marc},
      title = {Calculs de groupes d'homotopie stables de la sphère, par la suite spectrale d'{A}dams-{N}ovikov},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {185},
      year = {1984},
      number = {1},
      pages = {45--91},
      issn = {0025-5874},
      mrclass = {55Q45 (55T15)},
      mrnumber = {0724045},
      mrreviewer = {J. F. Adams},
      doi = {10.1007/BF01214973},
      url = {http://dx.doi.org/10.1007/BF01214973},
      zblnumber = {0509.55009},
      }
  • [Bau] Go to document T. Bauer, "Computation of the homotopy of the spectrum \tt tmf," in Groups, Homotopy and Configuration Spaces, Geom. Topol. Publ., Coventry, 2008, vol. 13, pp. 11-40.
    @INCOLLECTION{Bau,
      author = {Bauer, Tilman},
      title = {Computation of the homotopy of the spectrum {\tt tmf}},
      booktitle = {Groups, Homotopy and Configuration Spaces},
      series = {Geom. Topol. Monogr.},
      volume = {13},
      pages = {11--40},
      publisher = {Geom. Topol. Publ., Coventry},
      year = {2008},
      mrclass = {55N34 (55T15)},
      mrnumber = {2508200},
      mrreviewer = {Tyler D. Lawson},
      doi = {10.2140/gtm.2008.13.11},
      url = {http://dx.doi.org/10.2140/gtm.2008.13.11},
      zblnumber = {1147.55005},
      }
  • [BG] Go to document J. C. Becker and D. H. Gottlieb, "The transfer map and fiber bundles," Topology, vol. 14, pp. 1-12, 1975.
    @ARTICLE{BG,
      author = {Becker, J. C. and Gottlieb, D. H.},
      title = {The transfer map and fiber bundles},
      journal = {Topology},
      fjournal = {Topology. An International Journal of Mathematics},
      volume = {14},
      year = {1975},
      pages = {1--12},
      issn = {0040-9383},
      mrclass = {55E50 (55F10)},
      mrnumber = {0377873},
      mrreviewer = {J. P. May},
      doi = {10.1016/0040-9383(75)90029-4},
      url = {http://dx.doi.org/10.1016/0040-9383(75)90029-4},
      zblnumber = {0306.55017},
      }
  • [BHHM] M. Behrens, M. Hill, M. Hopkins, and M. Mahowald, Exotic spheres detected by topological modular forms.
    @MISC{BHHM,
      author = {Behrens, Mark and Hill, Mike and Hopkins, Mike and Mahowald, Mark},
      title = {Exotic spheres detected by topological modular forms},
      note = {preprint},
      zblnumber = {},
      }
  • [BJM] Go to document M. G. Barratt, J. D. S. Jones, and M. E. Mahowald, "Relations amongst Toda brackets and the Kervaire invariant in dimension $62$," J. London Math. Soc. (2), vol. 30, iss. 3, pp. 533-550, 1984.
    @ARTICLE{BJM,
      author = {Barratt, M. G. and Jones, J. D. S. and Mahowald, M. E.},
      title = {Relations amongst {T}oda brackets and the {K}ervaire invariant in dimension {$62$}},
      journal = {J. London Math. Soc. (2)},
      fjournal = {Journal of the London Mathematical Society. Second Series},
      volume = {30},
      year = {1984},
      number = {3},
      pages = {533--550},
      issn = {0024-6107},
      mrclass = {55Q45 (55T15)},
      mrnumber = {0810962},
      mrreviewer = {Haynes R. Miller},
      doi = {10.1112/jlms/s2-30.3.533},
      url = {http://dx.doi.org/10.1112/jlms/s2-30.3.533},
      zblnumber = {0606.55010},
      }
  • [BJM2] Go to document M. G. Barratt, J. D. S. Jones, and M. E. Mahowald, "The Kervaire invariant problem," in Proceedings of the Northwestern Homotopy Theory Conference, 1983, pp. 9-22.
    @INPROCEEDINGS{BJM2,
      author = {Barratt, M. G. and Jones, J. D. S. and Mahowald, M. E.},
      title = {The {K}ervaire invariant problem},
      booktitle = {Proceedings of the {N}orthwestern {H}omotopy {T}heory {C}onference},
      venue = {{E}vanston, {I}ll., 1982},
      series = {Contemp. Math.},
      volume = {19},
      pages = {9--22},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1983},
      mrclass = {55Q45 (55T15)},
      mrnumber = {0711039},
      mrreviewer = {Frederick Cohen},
      doi = {10.1090/conm/019/711039},
      url = {http://dx.doi.org/10.1090/conm/019/711039},
      zblnumber = {0528.55010},
      }
  • [BMT] Go to document M. G. Barratt, M. E. Mahowald, and M. C. Tangora, "Some differentials in the Adams spectral sequence. II," Topology, vol. 9, pp. 309-316, 1970.
    @ARTICLE{BMT,
      author = {Barratt, M. G. and Mahowald, M. E. and Tangora, M. C.},
      title = {Some differentials in the {A}dams spectral sequence. {II}},
      journal = {Topology},
      fjournal = {Topology. An International Journal of Mathematics},
      volume = {9},
      year = {1970},
      pages = {309--316},
      issn = {0040-9383},
      mrclass = {55.52},
      mrnumber = {0266215},
      mrreviewer = {J. F. Adams},
      doi = {10.1016/0040-9383(70)90055-8},
      url = {http://dx.doi.org/10.1016/0040-9383(70)90055-8},
      zblnumber = {0213.24901},
      }
  • [Br1] Go to document R. Bruner, "A new differential in the Adams spectral sequence," Topology, vol. 23, iss. 3, pp. 271-276, 1984.
    @ARTICLE{Br1,
      author = {Bruner, Robert},
      title = {A new differential in the {A}dams spectral sequence},
      journal = {Topology},
      fjournal = {Topology. An International Journal of Mathematics},
      volume = {23},
      year = {1984},
      number = {3},
      pages = {271--276},
      issn = {0040-9383},
      mrclass = {55Q45 (55T15)},
      mrnumber = {0770563},
      mrreviewer = {Stanley Kochman},
      doi = {10.1016/0040-9383(84)90010-7},
      url = {http://dx.doi.org/10.1016/0040-9383(84)90010-7},
      zblnumber = {0565.55024},
      }
  • [Br2] R. R. Bruner, The cohomology of the mod 2 Steenrod algebra: a computer calculation.
    @MISC{Br2,
      author = {Bruner, Robert R.},
      title = {The cohomology of the mod 2 {S}teenrod algebra: a computer calculation},
      note={preprint},
      }
  • [Coh] Go to document J. M. Cohen, "The decomposition of stable homotopy," Ann. of Math. (2), vol. 87, pp. 305-320, 1968.
    @ARTICLE{Coh,
      author = {Cohen, Joel M.},
      title = {The decomposition of stable homotopy},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {87},
      year = {1968},
      pages = {305--320},
      issn = {0003-486X},
      mrclass = {55.45},
      mrnumber = {0231377},
      mrreviewer = {J. Stasheff},
      doi = {10.2307/1970586},
      url = {http://dx.doi.org/10.2307/1970586},
      zblnumber = {0162.55102},
      }
  • [Con] Go to document E. H. Connell, "A topological $H$-cobordism theorem for $n\geq 5$," Illinois J. Math., vol. 11, pp. 300-309, 1967.
    @ARTICLE{Con,
      author = {Connell, E. H.},
      title = {A topological {$H$}-cobordism theorem for {$n\geq 5$}},
      journal = {Illinois J. Math.},
      fjournal = {Illinois Journal of Mathematics},
      volume = {11},
      year = {1967},
      pages = {300--309},
      issn = {0019-2082},
      mrclass = {57.05},
      mrnumber = {0212808},
      mrreviewer = {E. H. Brown},
      url = {http://projecteuclid.org/euclid.ijm/1256054669},
      zblnumber = {0146.45201},
      }
  • [Fre] Go to document M. H. Freedman, "The topology of four-dimensional manifolds," J. Differential Geom., vol. 17, iss. 3, pp. 357-453, 1982.
    @ARTICLE{Fre,
      author = {Freedman, Michael Hartley},
      title = {The topology of four-dimensional manifolds},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {17},
      year = {1982},
      number = {3},
      pages = {357--453},
      issn = {0022-040X},
      mrclass = {57N12 (57R80 57R99)},
      mrnumber = {0679066},
      mrreviewer = {John J. Walsh},
      url = {http://projecteuclid.org/euclid.jdg/1214437136},
      zblnumber = {0528.57011},
      }
  • [Freu] Go to document H. Freudenthal, "Über die Klassen der Sphärenabbildungen. I. Große Dimensionen," Comp. Math., vol. 5, pp. 299-314, 1938.
    @ARTICLE{Freu,
      author = {Freudenthal, H.},
      title = {Über die Klassen der Sphärenabbildungen. {I. G}roße Dimensionen},
      journal = {Comp. Math.},
      volume = {5},
      year = {1938},
      pages = {299--314},
      zblnumber = {0018.17705},
      url = {http://www.numdam.org/item?id=CM_1938__5__299_0},
      mrnumber = {1556999},
     }
  • [Hen] A. Henriques, "The homotopy groups of $tmf$ and of its localizations," in Topological Mocular Forms, Amer. Math. Soc., Providence, RI, 2014, vol. 201, pp. 189-205.
    @INCOLLECTION{Hen,
      author = {Henriques, Andre},
      title = {The homotopy groups of $tmf$ and of its localizations},
      booktitle = {Topological {M}ocular {F}orms},
      series={Math. Surveys Monogr.},
      VOLUME={201},
      note = {based on the Talbot Workshop, North Conway, NH, USA, March 25--31, 2007},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2014},
      pages = {189--205},
      mrnumber = {3223024},
      zblnumber = {1328.55015},
      }
  • [HHR] Go to document M. A. Hill, M. J. Hopkins, and D. C. Ravenel, "On the nonexistence of elements of Kervaire invariant one," Ann. of Math. (2), vol. 184, iss. 1, pp. 1-262, 2016.
    @ARTICLE{HHR,
      author = {Hill, M. A. and Hopkins, M. J. and Ravenel, D. C.},
      title = {On the nonexistence of elements of {K}ervaire invariant one},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {184},
      year = {2016},
      number = {1},
      pages = {1--262},
      issn = {0003-486X},
      mrclass = {55P91 (55N22 55P42 55Q45 55T15 55U35 57R15)},
      mrnumber = {3505179},
      mrreviewer = {Paul G. Goerss},
      doi = {10.4007/annals.2016.184.1.1},
      url = {http://dx.doi.org/10.4007/annals.2016.184.1.1},
      zblnumber = {06605831},
      }
  • [Hop] Go to document H. Hopf, "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche," Math. Ann., vol. 104, iss. 1, pp. 637-665, 1931.
    @ARTICLE{Hop,
      author = {Hopf, Heinz},
      title = {Über die {A}bbildungen der dreidimensionalen {S}phäre auf die {K}ugelfläche},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {104},
      year = {1931},
      number = {1},
      pages = {637--665},
      issn = {0025-5831},
      mrclass = {DML},
      mrnumber = {1512691},
      doi = {10.1007/BF01457962},
      url = {http://dx.doi.org/10.1007/BF01457962},
      zblnumber = {0001.40703},
      }
  • [Isa] D. C. Isaksen, Stable stems, 2014.
    @MISC{Isa,
      author = {Isaksen, Daniel C.},
      title = {Stable stems},
      arxiv = {1407.8418},
      year = {2014},
      }
  • [Isa2] D. C. Isaksen, Classical and motivic Adams charts, 2014.
    @MISC{Isa2,
      author = {Isaksen, Daniel C.},
      title = {Classical and motivic {A}dams charts},
      arxiv = {1401.4983},
      year = {2014},
      zblnumber = {},
      }
  • [IX] Go to document D. C. Isaksen and Z. Xu, "Motivic stable homotopy and the stable 51 and 52 stems," Topology Appl., vol. 190, pp. 31-34, 2015.
    @ARTICLE{IX,
      author = {Isaksen, Daniel C. and Xu, Zhouli},
      title = {Motivic stable homotopy and the stable 51 and 52 stems},
      journal = {Topology Appl.},
      volume = {190},
      year = {2015},
      pages = {31--34},
      issn = {0166-8641},
      mrclass = {55T15 (55Q45)},
      mrnumber = {3349503},
      mrreviewer = {Xiugui Liu},
      doi = {10.1016/j.topol.2015.04.008},
      url = {http://dx.doi.org/10.1016/j.topol.2015.04.008},
      zblnumber = {1327.55007},
      }
  • [KM2] Go to document M. A. Kervaire and J. W. Milnor, "Groups of homotopy spheres. I," Ann. of Math. (2), vol. 77, pp. 504-537, 1963.
    @ARTICLE{KM2,
      author = {Kervaire, Michel A. and Milnor, John W.},
      title = {Groups of homotopy spheres. {I}},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {77},
      year = {1963},
      pages = {504--537},
      issn = {0003-486X},
      mrclass = {57.10},
      mrnumber = {0148075},
      mrreviewer = {J. F. Adams},
      doi = {10.2307/1970128},
      url = {http://dx.doi.org/10.2307/1970128},
      zblnumber = {0115.40505},
      }
  • [KM] Go to document S. O. Kochman and M. E. Mahowald, "On the computation of stable stems," in The \vCech Centennial, Amer. Math. Soc., Providence, RI, 1995, vol. 181, pp. 299-316.
    @INCOLLECTION{KM,
      author = {Kochman, Stanley O. and Mahowald, Mark E.},
      title = {On the computation of stable stems},
      booktitle = {The \v{C}ech Centennial},
      venue = {{B}oston, {MA},
      1993},
      series = {Contemp. Math.},
      volume = {181},
      pages = {299--316},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1995},
      mrclass = {55Q45 (55T15)},
      mrnumber = {1320997},
      doi = {10.1090/conm/181/02039},
      url = {http://dx.doi.org/10.1090/conm/181/02039},
      zblnumber = {0819.55006},
      }
  • [Koc] Go to document S. O. Kochman, Stable Homotopy Groups of Spheres, Springer-Verlag, New York, 1990, vol. 1423.
    @BOOK{Koc,
      author = {Kochman, Stanley O.},
      title = {Stable Homotopy Groups of Spheres},
      series = {Lecture Notes in Math.},
      volume = {1423},
      titlenote = {A computer-assisted approach},
      publisher = {Springer-Verlag, New York},
      year = {1990},
      pages = {viii+330},
      isbn = {3-540-52468-1},
      mrclass = {55Q45 (55Q50 55T15)},
      mrnumber = {1052407},
      mrreviewer = {M. Mahowald},
      doi = {10.1007/BFb0083795},
      url = {http://dx.doi.org/10.1007/BFb0083795},
      zblnumber = {0688.55010},
      }
  • [KP] Go to document D. S. Kahn and S. B. Priddy, "The transfer and stable homotopy theory," Math. Proc. Cambridge Philos. Soc., vol. 83, iss. 1, pp. 103-111, 1978.
    @ARTICLE{KP,
      author = {Kahn, Daniel S. and Priddy, Stewart B.},
      title = {The transfer and stable homotopy theory},
      journal = {Math. Proc. Cambridge Philos. Soc.},
      fjournal = {Mathematical Proceedings of the Cambridge Philosophical Society},
      volume = {83},
      year = {1978},
      number = {1},
      pages = {103--111},
      issn = {0305-0041},
      mrclass = {55E45},
      mrnumber = {0464230},
      mrreviewer = {J. P. May},
      doi = {10.1017/S0305004100054335},
      url = {http://dx.doi.org/10.1017/S0305004100054335},
      zblnumber = {0373.55011},
      }
  • [Lin] Go to document H. Wên. Lin, "Algebraic Kahn-Priddy theorem," Pacific J. Math., vol. 96, iss. 2, pp. 435-455, 1981.
    @ARTICLE{Lin,
      author = {Lin, Wên Hsiung},
      title = {Algebraic {K}ahn-{P}riddy theorem},
      journal = {Pacific J. Math.},
      fjournal = {Pacific Journal of Mathematics},
      volume = {96},
      year = {1981},
      number = {2},
      pages = {435--455},
      issn = {0030-8730},
      mrclass = {55T15},
      mrnumber = {0637982},
      mrreviewer = {J. F. Adams},
      url = {http://projecteuclid.org/euclid.pjm/1102734795},
      zblnumber = {0504.55016},
      }
  • [Mah] M. Mahowald, "The order of the image of the $J$-homomorphism," in Proc. Advanced Study Inst. on Algebraic Topology, Vol. II, Mat. Inst., Aarhus Univ., Aarhus, 1970, pp. 376-384.
    @INCOLLECTION{Mah,
      author = {Mahowald, Mark},
      title = {The order of the image of the {$J$}-homomorphism},
      booktitle = {Proc. {A}dvanced {S}tudy {I}nst. on {A}lgebraic {T}opology, {V}ol. {II}},
      venue = {{A}arhus, 1970},
      pages = {376--384},
      publisher = {Mat. Inst., Aarhus Univ., Aarhus},
      year = {1970},
      mrclass = {55.40},
      mrnumber = {0276962},
      mrreviewer = {J. F. Adams},
      zblnumber = {0224.55019},
      }
  • [May] Go to document P. J. May, "Matric Massey products," J. Algebra, vol. 12, pp. 533-568, 1969.
    @ARTICLE{May,
      author = {May, J. Peter},
      title = {Matric {M}assey products},
      journal = {J. Algebra},
      fjournal = {Journal of Algebra},
      volume = {12},
      year = {1969},
      pages = {533--568},
      issn = {0021-8693},
      mrclass = {18.20 (55.00)},
      mrnumber = {0238929},
      mrreviewer = {R. E. Mosher},
      doi = {10.1016/0021-8693(69)90027-1},
      url = {http://dx.doi.org/10.1016/0021-8693(69)90027-1},
      zblnumber = {0192.34302},
      }
  • [May2] Go to document P. J. May, The Cohomology of Restricted Lie Algebras and of Hopf Algebras: Application to the Steenrod Algegra, ProQuest LLC, Ann Arbor, MI, 1964.
    @BOOK{May2,
      author = {May, J. Peter},
      title = {The Cohomology of Restricted Lie Algebras and of Hopf Algebras: Application to the Steenrod Algegra},
      note = {thesis (Ph.D.)--Princeton University},
      publisher = {ProQuest LLC, Ann Arbor, MI},
      year = {1964},
      pages = {171},
      mrclass = {Thesis},
      mrnumber = {2614527},
      url = {https://search.proquest.com/docview/302273947?accountid=13314},
      zblnumber = {},
      }
  • [Mil] Go to document J. Milnor, "On manifolds homeomorphic to the $7$-sphere," Ann. of Math. (2), vol. 64, pp. 399-405, 1956.
    @ARTICLE{Mil,
      author = {Milnor, John},
      title = {On manifolds homeomorphic to the {$7$}-sphere},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {64},
      year = {1956},
      pages = {399--405},
      issn = {0003-486X},
      mrclass = {55.0X},
      mrnumber = {0082103},
      mrreviewer = {J. C. Moore},
      doi = {10.2307/1969983},
      url = {http://dx.doi.org/10.2307/1969983},
      zblnumber = {0072.18402},
      }
  • [Mil2] Go to document J. Milnor, "Differential topology forty-six years later," Notices Amer. Math. Soc., vol. 58, iss. 6, pp. 804-809, 2011.
    @ARTICLE{Mil2,
      author = {Milnor, John},
      title = {Differential topology forty-six years later},
      journal = {Notices Amer. Math. Soc.},
      fjournal = {Notices of the American Mathematical Society},
      volume = {58},
      year = {2011},
      number = {6},
      pages = {804--809},
      issn = {0002-9920},
      mrclass = {57-03 (01A60)},
      mrnumber = {2839925},
      mrreviewer = {Laurence R. Taylor},
      zblnumber = {1225.01040},
      URL = {http://www.ams.org/notices/201106/rtx110600804p.pdf},
     }
  • [Moi] Go to document E. E. Moise, "Affine structures in $3$-manifolds. V. The triangulation theorem and Hauptvermutung," Ann. of Math. (2), vol. 56, pp. 96-114, 1952.
    @ARTICLE{Moi,
      author = {Moise, Edwin E.},
      title = {Affine structures in {$3$}-manifolds. {V}. {T}he triangulation theorem and {H}auptvermutung},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {56},
      year = {1952},
      pages = {96--114},
      issn = {0003-486X},
      mrclass = {56.0X},
      mrnumber = {0048805},
      mrreviewer = {S. S. Cairns},
      doi = {10.2307/1969769},
      url = {http://dx.doi.org/10.2307/1969769},
      zblnumber = {0048.17102},
      }
  • [Mos] Go to document M. R. F. Moss, "Secondary compositions and the Adams spectral sequence," Math. Z., vol. 115, pp. 283-310, 1970.
    @ARTICLE{Mos,
      author = {Moss, R. Michael F.},
      title = {Secondary compositions and the {A}dams spectral sequence},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {115},
      year = {1970},
      pages = {283--310},
      issn = {0025-5874},
      mrclass = {55.52},
      mrnumber = {0266216},
      mrreviewer = {J. P. May},
      doi = {10.1007/BF01129978},
      url = {http://dx.doi.org/10.1007/BF01129978},
      zblnumber = {0188.28501},
      }
  • [MT] Go to document M. Mahowald and M. Tangora, "Some differentials in the Adams spectral sequence," Topology, vol. 6, pp. 349-369, 1967.
    @ARTICLE{MT,
      author = {Mahowald, Mark and Tangora, Martin},
      title = {Some differentials in the {A}dams spectral sequence},
      journal = {Topology},
      fjournal = {Topology. An International Journal of Mathematics},
      volume = {6},
      year = {1967},
      pages = {349--369},
      issn = {0040-9383},
      mrclass = {55.52},
      mrnumber = {0214072},
      mrreviewer = {J. F. Adams},
      doi = {10.1016/0040-9383(67)90023-7},
      url = {http://dx.doi.org/10.1016/0040-9383(67)90023-7},
      zblnumber = {0166.19004},
      }
  • [MT2] Go to document M. Mimura and H. Toda, "The $(n+20)$-th homotopy groups of $n$-spheres," J. Math. Kyoto Univ., vol. 3, pp. 37-58, 1963.
    @ARTICLE{MT2,
      author = {Mimura, Mamoru and Toda, Hirosi},
      title = {The {$(n+20)$}-th homotopy groups of {$n$}-spheres},
      journal = {J. Math. Kyoto Univ.},
      fjournal = {Journal of Mathematics of Kyoto University},
      volume = {3},
      year = {1963},
      pages = {37--58},
      issn = {0023-608X},
      mrclass = {55.45},
      mrnumber = {0157384},
      mrreviewer = {R. Bott},
      doi = {10.1215/kjm/1250524854},
      url = {http://dx.doi.org/10.1215/kjm/1250524854},
      zblnumber = {0129.15403},
      }
  • [Nak] O. Nakamura, "Some differentials in the ${ mod} 3$ Adams spectral sequence," Bull. Sci. Engrg. Div. Univ. Ryukyus Math. Natur. Sci., iss. 19, pp. 1-25, 1975.
    @ARTICLE{Nak,
      author = {Nakamura, Osamu},
      title = {Some differentials in the {${\rm mod} 3$} {A}dams spectral sequence},
      journal = {Bull. Sci. Engrg. Div. Univ. Ryukyus Math. Natur. Sci.},
      number = {19},
      year = {1975},
      pages = {1--25},
      mrclass = {55E45},
      mrnumber = {0385852},
      mrreviewer = {Martin C. Tangora},
      zblnumber = {0368.55017},
      }
  • [New] Go to document M. H. A. Newman, "The engulfing theorem for topological manifolds," Ann. of Math. (2), vol. 84, pp. 555-571, 1966.
    @ARTICLE{New,
      author = {Newman, M. H. A.},
      title = {The engulfing theorem for topological manifolds},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {84},
      year = {1966},
      pages = {555--571},
      issn = {0003-486X},
      mrclass = {54.78},
      mrnumber = {0203708},
      mrreviewer = {R. H. Rosen},
      doi = {10.2307/1970460},
      url = {http://dx.doi.org/10.2307/1970460},
      zblnumber = {0166.19801},
      }
  • [Per] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, 2002.
    @MISC{Per,
      author = {Perelman, Grisha},
      title = {The entropy formula for the {R}icci flow and its geometric applications},
      arxiv = {math/0211159},
      year = {2002},
      zblnumber = {1130.53001},
      }
  • [Pon] L. S. Pontryagin, "Homotopy classification of the mappings of an $(n+2)$-dimensional sphere on an $n$-dimensional one," Doklady Akad. Nauk SSSR (N.S.), vol. 70, pp. 957-959, 1950.
    @ARTICLE{Pon,
      author = {Pontryagin, L. S.},
      title = {Homotopy classification of the mappings of an {$(n+2)$}-dimensional sphere on an {$n$}-dimensional one},
      journal = {Doklady Akad. Nauk SSSR (N.S.)},
      volume = {70},
      year = {1950},
      pages = {957--959},
      mrclass = {56.0X},
      mrnumber = {0042121},
      mrreviewer = {B. Eckmann},
      zblnumber = {0035.11101},
      }
  • [Pri] Go to document S. B. Priddy, "Koszul resolutions," Trans. Amer. Math. Soc., vol. 152, pp. 39-60, 1970.
    @ARTICLE{Pri,
      author = {Priddy, Stewart B.},
      title = {Koszul resolutions},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the American Mathematical Society},
      volume = {152},
      year = {1970},
      pages = {39--60},
      issn = {0002-9947},
      mrclass = {18.20},
      mrnumber = {0265437},
      mrreviewer = {A. K. Bousfield},
      doi = {10.2307/1995637},
      url = {http://dx.doi.org/10.2307/1995637},
      zblnumber = {0261.18016},
      }
  • [Qui] Go to document D. Quillen, "The Adams conjecture," Topology, vol. 10, pp. 67-80, 1971.
    @ARTICLE{Qui,
      author = {Quillen, Daniel},
      title = {The {A}dams conjecture},
      journal = {Topology},
      fjournal = {Topology. An International Journal of Mathematics},
      volume = {10},
      year = {1971},
      pages = {67--80},
      issn = {0040-9383},
      mrclass = {55.30},
      mrnumber = {0279804},
      mrreviewer = {J. F. Adams},
      doi = {10.1016/0040-9383(71)90018-8},
      url = {http://dx.doi.org/10.1016/0040-9383(71)90018-8},
      zblnumber = {0219.55013},
      }
  • [Rav] D. C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Academic Press, Inc., Orlando, FL, 1986, vol. 121.
    @BOOK{Rav,
      author = {Ravenel, Douglas C.},
      title = {Complex Cobordism and Stable Homotopy Groups of Spheres},
      series = {Pure Appl. Math.},
      volume = {121},
      publisher = {Academic Press, Inc., Orlando, FL},
      year = {1986},
      pages = {xx+413},
      isbn = {0-12-583430-6; 0-12-583431-4},
      mrclass = {55-02 (55Qxx 57-02)},
      mrnumber = {0860042},
      mrreviewer = {Joseph Neisendorfer},
      zblnumber = {0608.55001},
      }
  • [Rok] V. A. Rohlin, "On a mapping of the $(n+3)$-dimensional sphere into the $n$-dimensional sphere," Doklady Akad. Nauk SSSR (N.S.), vol. 80, pp. 541-544, 1951.
    @ARTICLE{Rok,
      author = {Rohlin, V. A.},
      title = {On a mapping of the {$(n+3)$}-dimensional sphere into the {$n$}-dimensional sphere},
      journal = {Doklady Akad. Nauk SSSR (N.S.)},
      volume = {80},
      year = {1951},
      pages = {541--544},
      mrclass = {56.0X},
      mrnumber = {0046042},
      mrreviewer = {P. J. Hilton},
      zblnumber = {},
      }
  • [Ser] Go to document . J-P. Serre, "Homologie singulière des espaces fibrés. Applications," Ann. of Math. (2), vol. 54, pp. 425-505, 1951.
    @ARTICLE{Ser,
      author = {Serre, {\relax J-P}},
      title = {Homologie singulière des espaces fibrés. {A}pplications},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {54},
      year = {1951},
      pages = {425--505},
      issn = {0003-486X},
      mrclass = {56.0X},
      mrnumber = {0045386},
      mrreviewer = {W. S. Massey},
      doi = {10.2307/1969485},
      url = {http://dx.doi.org/10.2307/1969485},
      zblnumber = {0045.26003},
      }
  • [Sma] Go to document S. Smale, "Generalized Poincaré’s conjecture in dimensions greater than four," Ann. of Math. (2), vol. 74, pp. 391-406, 1961.
    @ARTICLE{Sma,
      author = {Smale, Stephen},
      title = {Generalized {P}oincaré's conjecture in dimensions greater than four},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {74},
      year = {1961},
      pages = {391--406},
      issn = {0003-486X},
      mrclass = {57.01 (57.10)},
      mrnumber = {0137124},
      mrreviewer = {M. W. Hirsch},
      doi = {10.2307/1970239},
      url = {http://dx.doi.org/10.2307/1970239},
      zblnumber = {0099.39202},
      }
  • [Sul] Go to document D. Sullivan, "Genetics of homotopy theory and the Adams conjecture," Ann. of Math. (2), vol. 100, pp. 1-79, 1974.
    @ARTICLE{Sul,
      author = {Sullivan, Dennis},
      title = {Genetics of homotopy theory and the {A}dams conjecture},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {100},
      year = {1974},
      pages = {1--79},
      issn = {0003-486X},
      mrclass = {55F25 (55D15 57F99)},
      mrnumber = {0442930},
      mrreviewer = {D. B. Fuks},
      doi = {10.2307/1970841},
      url = {http://dx.doi.org/10.2307/1970841},
      zblnumber = {0355.57007},
      }
  • [Tan1] Go to document M. C. Tangora, "On the cohomology of the Steenrod algebra," Math. Z., vol. 116, pp. 18-64, 1970.
    @ARTICLE{Tan1,
      author = {Tangora, Martin C.},
      title = {On the cohomology of the {S}teenrod algebra},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {116},
      year = {1970},
      pages = {18--64},
      issn = {0025-5874},
      mrclass = {55.34},
      mrnumber = {0266205},
      mrreviewer = {Larry Smith},
      doi = {10.1007/BF01110185},
      url = {http://dx.doi.org/10.1007/BF01110185},
      zblnumber = {0198.28202},
      }
  • [Tan2] M. Tangora, "Some extension questions in the Adams spectral sequence," in Proceedings of the Advanced Study Institute on the Algebraic Topology, Vol.III, Mat. Inst., Aarhus Univ., Aarhus, 1970, vol. 13, pp. 578-587.
    @incollection{Tan2,
      author = {Tangora, Martin},
      title = {Some extension questions in the {A}dams spectral sequence},
      booktitle = {Proceedings of the {A}dvanced {S}tudy {I}nstitute on the {A}lgebraic {T}opology, {V}ol.{III}},
      VENUE={{A}arhus {U}niv., {A}arhus, 1970},
      pages = {578--587},
      SERIES={Various Publ. Ser.},
      VOLUME={13},
      publisher = {Mat. Inst., Aarhus Univ., Aarhus},
      year = {1970},
      mrclass = {55E45},
      mrnumber = {0339163},
      mrreviewer = {R. M. F. Moss},
      zblnumber = {0225.55018},
      }
  • [Tan3] Go to document M. C. Tangora, "Computing the homology of the lambda algebra," Mem. Amer. Math. Soc., vol. 58, iss. 337, p. v, 1985.
    @ARTICLE{Tan3,
      author = {Tangora, Martin C.},
      title = {Computing the homology of the lambda algebra},
      journal = {Mem. Amer. Math. Soc.},
      fjournal = {Memoirs of the American Mathematical Society},
      volume = {58},
      year = {1985},
      number = {337},
      pages = {v+163},
      issn = {0065-9266},
      mrclass = {55T15 (18-04 55-04 55Q40)},
      mrnumber = {0818916},
      doi = {10.1090/memo/0337},
      url = {http://dx.doi.org/10.1090/memo/0337},
      zblnumber = {0584.55019},
      }
  • [Tan4] M. Tangora, "Some homotopy groups mod $3$," in Conference on Homotopy Theory, Soc. Mat. Mexicana, México, 1975, vol. 1, pp. 227-245.
    @INCOLLECTION{Tan4,
      author = {Tangora, Martin},
      title = {Some homotopy groups mod {$3$}},
      booktitle = {Conference on Homotopy Theory},
      venue = {{E}vanston, {I}ll., 1974},
      series = {Notas Mat. Simpos.},
      volume = {1},
      pages = {227--245},
      publisher = {Soc. Mat. Mexicana, México},
      year = {1975},
      mrclass = {55Q45},
      mrnumber = {0761731},
      zblnumber = {0334.55014},
      }
  • [Tod] H. Toda, Composition Methods in Homotopy Groups of Spheres, Princeton Univ. Press, Princeton, NJ, 1962, vol. 49.
    @BOOK{Tod,
      author = {Toda, Hirosi},
      title = {Composition Methods in Homotopy Groups of Spheres},
      series = {Ann. of Math. Stud.},
      volume = {49},
      publisher = {Princeton Univ. Press, Princeton, NJ},
      year = {1962},
      pages = {v+193},
      mrclass = {55.45},
      mrnumber = {0143217},
      mrreviewer = {J. F. Adams},
      zblnumber = {0101.40703},
      }
  • [Whi] Go to document G. W. Whitehead, "The $(n+2)^{ nd}$ homotopy group of the $n$-sphere," Ann. of Math. (2), vol. 52, pp. 245-247, 1950.
    @ARTICLE{Whi,
      author = {Whitehead, George W.},
      title = {The {$(n+2)^{\rm nd}$} homotopy group of the {$n$}-sphere},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {52},
      year = {1950},
      pages = {245--247},
      issn = {0003-486X},
      mrclass = {56.0X},
      mrnumber = {0037507},
      mrreviewer = {H. Freudenthal},
      doi = {10.2307/1969466},
      url = {http://dx.doi.org/10.2307/1969466},
      zblnumber = {0037.39703},
      }
  • [WX] G. Wang and Z. Xu, The algebraic Atiyah-Hirzebruch spectral sequence of real projective spectra, 2016.
    @MISC{WX,
      author = {Wang, Guozhen and Xu, Zhouli},
      title = {The algebraic {A}tiyah-{H}irzebruch spectral sequence of real projective spectra},
      arxiv = {1601.02185},
      year = {2016},
      zblnumber = {},
      }
  • [Xu] Go to document Z. Xu, "The strong Kervaire invariant problem in dimension 62," Geom. Topol., vol. 20, iss. 3, pp. 1611-1624, 2016.
    @ARTICLE{Xu,
      author = {Xu, Zhouli},
      title = {The strong {K}ervaire invariant problem in dimension 62},
      journal = {Geom. Topol.},
      fjournal = {Geometry \& Topology},
      volume = {20},
      year = {2016},
      number = {3},
      pages = {1611--1624},
      issn = {1465-3060},
      mrclass = {55Q45},
      mrnumber = {3523064},
      mrreviewer = {Katsumi Shimomura},
      doi = {10.2140/gt.2016.20.1611},
      url = {http://dx.doi.org/10.2140/gt.2016.20.1611},
      zblnumber = {1352.55007},
      }

Authors

Guozhen Wang

Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China, 200433 and Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark

Zhouli Xu

Department of Mathematics, The University of Chicago, Chicago, IL

Current address:

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA