Highly connected $7$-manifolds and non-negative sectional curvature

Abstract

In this article, a six-parameter family of highly connected 7-manifolds which admit an $\mathrm {SO}(3)$-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an $\mathrm{SO}(3)$-invariant metric of non-negative curvature.

  • [At] Go to document M. F. Atiyah, "Characters and cohomology of finite groups," Inst. Hautes Études Sci. Publ. Math., iss. 9, pp. 23-64, 1961.
    @ARTICLE{At,
      author = {Atiyah, M. F.},
      title = {Characters and cohomology of finite groups},
      journal = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
      fjournal = {Institut des Hautes \'{E}tudes Scientifiques. Publications Mathématiques},
      number = {9},
      year = {1961},
      pages = {23--64},
      issn = {0073-8301},
      mrclass = {18.20},
      mrnumber = {0148722},
      mrreviewer = {F. Hirzebruch},
      url = {http://www.numdam.org/item?id=PMIHES_1961__9__23_0},
      zblnumber = {0107.02303},
      }
  • [APS] Go to document M. F. Atiyah, V. K. Patodi, and I. M. Singer, "Spectral asymmetry and Riemannian geometry. II," Math. Proc. Cambridge Philos. Soc., vol. 78, iss. 3, pp. 405-432, 1975.
    @ARTICLE{APS,
      author = {Atiyah, M. F. and Patodi, V. K. and Singer, I. M.},
      title = {Spectral asymmetry and {R}iemannian geometry. {II}},
      journal = {Math. Proc. Cambridge Philos. Soc.},
      fjournal = {Mathematical Proceedings of the Cambridge Philosophical Society},
      volume = {78},
      year = {1975},
      number = {3},
      pages = {405--432},
      issn = {0305-0041},
      mrclass = {58G10 (57D85 57E15)},
      mrnumber = {0397798},
      mrreviewer = {Kh. Knapp},
      doi = {10.1017/S0305004100051872},
      url = {https://doi.org/10.1017/S0305004100051872},
      zblnumber = {0314.58016},
      }
  • [Ba] Go to document D. Barden, "Simply connected five-manifolds," Ann. of Math. (2), vol. 82, pp. 365-385, 1965.
    @ARTICLE{Ba,
      author = {Barden, D.},
      title = {Simply connected five-manifolds},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {82},
      year = {1965},
      pages = {365--385},
      issn = {0003-486X},
      mrclass = {57.10},
      mrnumber = {0184241},
      mrreviewer = {S. Smale},
      doi = {10.2307/1970702},
      url = {https://doi.org/10.2307/1970702},
      zblnumber = {0136.20602},
      }
  • [BGV] N. Berline, E. Getzler, and M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag, Berlin, 1992, vol. 298.
    @BOOK{BGV,
      author = {Berline, Nicole and Getzler, Ezra and Vergne, Michèle},
      title = {Heat Kernels and {D}irac Operators},
      series = {Grundlehren Math. Wiss.},
      volume = {298},
      publisher = {Springer-Verlag, Berlin},
      year = {1992},
      pages = {viii+369},
      isbn = {3-540-53340-0},
      mrclass = {58G10 (58G11)},
      mrnumber = {1215720},
      mrreviewer = {Alejandro Uribe},
      zblnumber = {0744.58001},
      }
  • [BC1] Go to document J. Bismut and J. Cheeger, "$\eta$-invariants and their adiabatic limits," J. Amer. Math. Soc., vol. 2, iss. 1, pp. 33-70, 1989.
    @ARTICLE{BC1,
      author = {Bismut, Jean-Michel and Cheeger, Jeff},
      title = {{$\eta$}-invariants and their adiabatic limits},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the Amer. Math. Soc.},
      volume = {2},
      year = {1989},
      number = {1},
      pages = {33--70},
      issn = {0894-0347},
      mrclass = {58G10 (58C50 58G12 58G20)},
      mrnumber = {0966608},
      mrreviewer = {Jürgen Eichhorn},
      doi = {10.2307/1990912},
      url = {https://doi.org/10.2307/1990912},
      zblnumber = {0671.58037},
      }
  • [BF] Go to document J. Bismut and D. S. Freed, "The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem," Comm. Math. Phys., vol. 107, iss. 1, pp. 103-163, 1986.
    @ARTICLE{BF,
      author = {Bismut, Jean-Michel and Freed, Daniel S.},
      title = {The analysis of elliptic families. {II}. {D}irac operators, eta invariants, and the holonomy theorem},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {107},
      year = {1986},
      number = {1},
      pages = {103--163},
      issn = {0010-3616},
      mrclass = {58G11 (58G10 58G32)},
      mrnumber = {0861886},
      mrreviewer = {Ezra Getzler},
      doi = {10.1007/BF01206955},
      url = {https://doi.org/10.1007/BF01206955},
      zblnumber = {0657.58038},
      }
  • [BL] Go to document J. Bismut and G. Lebeau, "Complex immersions and Quillen metrics," Inst. Hautes Études Sci. Publ. Math., iss. 74, p. ii, 1991.
    @ARTICLE{BL,
      author = {Bismut, Jean-Michel and Lebeau, Gilles},
      title = {Complex immersions and {Q}uillen metrics},
      journal = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
      fjournal = {Institut des Hautes \'{E}tudes Scientifiques. Publications Mathématiques},
      number = {74},
      year = {1991},
      pages = {ii+298 pp. (1992)},
      issn = {0073-8301},
      mrclass = {58G26 (32L10)},
      mrnumber = {1188532},
      mrreviewer = {Jürgen Eichhorn},
      url = {http://www.numdam.org/item?id=PMIHES_1991__74__298_0},
      zblnumber = {0784.32010},
      }
  • [BH] Go to document A. Borel and F. Hirzebruch, "Characteristic classes and homogeneous spaces. III," Amer. J. Math., vol. 82, pp. 491-504, 1960.
    @ARTICLE{BH,
      author = {Borel, A. and Hirzebruch, F.},
      title = {Characteristic classes and homogeneous spaces. {III}},
      journal = {Amer. J. Math.},
      fjournal = {American Journal of Mathematics},
      volume = {82},
      year = {1960},
      pages = {491--504},
      issn = {0002-9327},
      mrclass = {57.00},
      mrnumber = {0120664},
      mrreviewer = {R. Bott},
      doi = {10.2307/2372969},
      url = {https://doi.org/10.2307/2372969},
      zblnumber = {0097.36401},
      }
  • [Cr] Go to document D. J. Crowley, The classification of highly connected manifolds in dimensions 7 and 15, ProQuest LLC, Ann Arbor, MI, 2002.
    @BOOK{Cr,
      author = {Crowley, Diarmuid John},
      title = {The classification of highly connected manifolds in dimensions 7 and 15},
      note = {thesis (Ph.D.), Indiana Univ.},
      publisher = {ProQuest LLC, Ann Arbor, MI},
      year = {2002},
      pages = {151},
      isbn = {978-0493-69427-6},
      mrclass = {Thesis},
      mrnumber = {2703475},
      url = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3054376},
      zblnumber = {0999.57027},
      }
  • [CE] Go to document D. Crowley and C. M. Escher, "A classification of $S^3$-bundles over $S^4$," Differential Geom. Appl., vol. 18, iss. 3, pp. 363-380, 2003.
    @ARTICLE{CE,
      author = {Crowley, Diarmuid and Escher, Christine M.},
      title = {A classification of {$S^3$}-bundles over {$S^4$}},
      journal = {Differential Geom. Appl.},
      fjournal = {Differential Geometry and its Applications},
      volume = {18},
      year = {2003},
      number = {3},
      pages = {363--380},
      issn = {0926-2245},
      mrclass = {55R15 (53C21 55R40 57T35)},
      mrnumber = {1975035},
      mrreviewer = {Krishnan Shankar},
      doi = {10.1016/S0926-2245(03)00012-3},
      url = {https://doi.org/10.1016/S0926-2245(03)00012-3},
      zblnumber = {1027.55014},
      }
  • [CG] Go to document D. Crowley and S. Goette, "Kreck-Stolz invariants for quaternionic line bundles," Trans. Amer. Math. Soc., vol. 365, iss. 6, pp. 3193-3225, 2013.
    @ARTICLE{CG,
      author = {Crowley, Diarmuid and Goette, Sebastian},
      title = {Kreck-{S}tolz invariants for quaternionic line bundles},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the Amer. Math. Soc.},
      volume = {365},
      year = {2013},
      number = {6},
      pages = {3193--3225},
      issn = {0002-9947},
      mrclass = {58J28 (57R55)},
      mrnumber = {3034463},
      mrreviewer = {Man-Ho Ho},
      doi = {10.1090/S0002-9947-2012-05732-1},
      url = {https://doi.org/10.1090/S0002-9947-2012-05732-1},
      zblnumber = {1281.58014},
      }
  • [CN] Go to document D. Crowley and J. Nordström, "The classification of 2-connected 7-manifolds," Proc. Lond. Math. Soc. (3), vol. 119, iss. 1, pp. 1-54, 2019.
    @ARTICLE{CN,
      author = {Crowley, Diarmuid and Nordström, Johannes},
      title = {The classification of 2-connected 7-manifolds},
      journal = {Proc. Lond. Math. Soc. (3)},
      fjournal = {Proceedings of the London Mathematical Society. Third Series},
      volume = {119},
      year = {2019},
      number = {1},
      pages = {1--54},
      issn = {0024-6115},
      mrclass = {57R15 (57R50 57R65)},
      mrnumber = {3957830},
      doi = {10.1112/plms.12222},
      url = {https://doi.org/10.1112/plms.12222},
      zblnumber = {07094283},
      }
  • [Dai] Go to document X. Dai, "Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence," J. Amer. Math. Soc., vol. 4, iss. 2, pp. 265-321, 1991.
    @ARTICLE{Dai,
      author = {Dai, Xianzhe},
      title = {Adiabatic limits, nonmultiplicativity of signature, and {L}eray spectral sequence},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the Amer. Math. Soc.},
      volume = {4},
      year = {1991},
      number = {2},
      pages = {265--321},
      issn = {0894-0347},
      mrclass = {58G10 (55T10 58G25)},
      mrnumber = {1088332},
      mrreviewer = {Jürgen Eichhorn},
      doi = {10.2307/2939276},
      url = {https://doi.org/10.2307/2939276},
      zblnumber = {0736.58039},
      }
  • [De] Go to document O. Dearricott, "A 7-manifold with positive curvature," Duke Math. J., vol. 158, iss. 2, pp. 307-346, 2011.
    @ARTICLE{De,
      author = {Dearricott, Owen},
      title = {A 7-manifold with positive curvature},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {158},
      year = {2011},
      number = {2},
      pages = {307--346},
      issn = {0012-7094},
      mrclass = {53C21 (53C25)},
      mrnumber = {2805071},
      mrreviewer = {David J. Wraith},
      doi = {10.1215/00127094-1334022},
      url = {https://doi.org/10.1215/00127094-1334022},
      zblnumber = {1221.53069},
      }
  • [DV] Go to document J. DeVito, "The classification of compact simply connected biquotients in dimensions 4 and 5," Differential Geom. Appl., vol. 34, pp. 128-138, 2014.
    @ARTICLE{DV,
      author = {DeVito, Jason},
      title = {The classification of compact simply connected biquotients in dimensions 4 and 5},
      journal = {Differential Geom. Appl.},
      fjournal = {Differential Geometry and its Applications},
      volume = {34},
      year = {2014},
      pages = {128--138},
      issn = {0926-2245},
      mrclass = {53C30 (57R55)},
      mrnumber = {3209541},
      mrreviewer = {Martin Kerin},
      doi = {10.1016/j.difgeo.2014.04.002},
      url = {https://doi.org/10.1016/j.difgeo.2014.04.002},
      zblnumber = {1298.53043},
      }
  • [Do1] Go to document H. Donnelly, "Spectral geometry and invariants from differential topology," Bull. London Math. Soc., vol. 7, pp. 147-150, 1975.
    @ARTICLE{Do1,
      author = {Donnelly, Harold},
      title = {Spectral geometry and invariants from differential topology},
      journal = {Bull. London Math. Soc.},
      fjournal = {The Bulletin of the London Mathematical Society},
      volume = {7},
      year = {1975},
      pages = {147--150},
      issn = {0024-6093},
      mrclass = {58G99 (57D55)},
      mrnumber = {0372929},
      mrreviewer = {J. Eells},
      doi = {10.1112/blms/7.2.147},
      url = {https://doi.org/10.1112/blms/7.2.147},
      zblnumber = {0306.58019},
      }
  • [Do2] Go to document H. Donnelly, "Eta invariants for $G$-spaces," Indiana Univ. Math. J., vol. 27, iss. 6, pp. 889-918, 1978.
    @ARTICLE{Do2,
      author = {Donnelly, Harold},
      title = {Eta invariants for {$G$}-spaces},
      journal = {Indiana Univ. Math. J.},
      fjournal = {Indiana Univ. Mathematics Journal},
      volume = {27},
      year = {1978},
      number = {6},
      pages = {889--918},
      issn = {0022-2518},
      mrclass = {58G25 (53C05 57R20 58G10)},
      mrnumber = {0511246},
      mrreviewer = {Akira Asada},
      doi = {10.1512/iumj.1978.27.27060},
      url = {https://doi.org/10.1512/iumj.1978.27.27060},
      zblnumber = {0402.58006},
      }
  • [DPR] Go to document C. Durán, T. Püttmann, and A. Rigas, "An infinite family of Gromoll-Meyer spheres," Arch. Math. (Basel), vol. 95, iss. 3, pp. 269-282, 2010.
    @ARTICLE{DPR,
      author = {Dur\'{a}n, Carlos and Püttmann, Thomas and Rigas, A.},
      title = {An infinite family of {G}romoll-{M}eyer spheres},
      journal = {Arch. Math. (Basel)},
      fjournal = {Archiv der Mathematik},
      volume = {95},
      year = {2010},
      number = {3},
      pages = {269--282},
      issn = {0003-889X},
      mrclass = {57R60 (53C22 57R55 57R91)},
      mrnumber = {2719385},
      mrreviewer = {David J. Wraith},
      doi = {10.1007/s00013-010-0161-x},
      url = {https://doi.org/10.1007/s00013-010-0161-x},
      zblnumber = {1258.53039},
      }
  • [EK] Go to document J. Eells Jr. and N. H. Kuiper, "An invariant for certain smooth manifolds," Ann. Mat. Pura Appl. (4), vol. 60, pp. 93-110, 1962.
    @ARTICLE{EK,
      author = {Eells, Jr., James and Kuiper, Nicolaas H.},
      title = {An invariant for certain smooth manifolds},
      journal = {Ann. Mat. Pura Appl. (4)},
      fjournal = {Annali di Matematica Pura ed Applicata. Serie Quarta},
      volume = {60},
      year = {1962},
      pages = {93--110},
      issn = {0003-4622},
      mrclass = {57.32 (57.10)},
      mrnumber = {0156356},
      mrreviewer = {R. H. Szczarba},
      doi = {10.1007/BF02412768},
      url = {https://doi.org/10.1007/BF02412768},
      zblnumber = {0119.18704},
      }
  • [Es] Go to document J. -H. Eschenburg, "Cohomology of biquotients," Manuscripta Math., vol. 75, iss. 2, pp. 151-166, 1992.
    @ARTICLE{Es,
      author = {Eschenburg, J.-H.},
      title = {Cohomology of biquotients},
      journal = {Manuscripta Math.},
      fjournal = {Manuscripta Mathematica},
      volume = {75},
      year = {1992},
      number = {2},
      pages = {151--166},
      issn = {0025-2611},
      mrclass = {57T15 (53C30 55R20)},
      mrnumber = {1160094},
      mrreviewer = {Samuel Evens},
      doi = {10.1007/BF02567078},
      url = {https://doi.org/10.1007/BF02567078},
      zblnumber = {0769.53029},
      }
  • [EsKe] Go to document J. -H. Eschenburg and M. Kerin, "Almost positive curvature on the Gromoll-Meyer sphere," Proc. Amer. Math. Soc., vol. 136, iss. 9, pp. 3263-3270, 2008.
    @ARTICLE{EsKe,
      author = {Eschenburg, J.-H. and Kerin, M.},
      title = {Almost positive curvature on the {G}romoll-{M}eyer sphere},
      journal = {Proc. Amer. Math. Soc.},
      fjournal = {Proceedings of the Amer. Math. Soc.},
      volume = {136},
      year = {2008},
      number = {9},
      pages = {3263--3270},
      issn = {0002-9939},
      mrclass = {53C21 (53C20 53C30 57R60)},
      mrnumber = {2407092},
      mrreviewer = {David J. Wraith},
      doi = {10.1090/S0002-9939-08-09429-X},
      url = {https://doi.org/10.1090/S0002-9939-08-09429-X},
      zblnumber = {1153.53023},
      }
  • [GGK] Go to document F. Galaz-Garcia and M. Kerin, "Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension," Math. Z., vol. 276, iss. 1-2, pp. 133-152, 2014.
    @ARTICLE{GGK,
      author = {Galaz-Garcia, Fernando and Kerin, Martin},
      title = {Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {276},
      year = {2014},
      number = {1-2},
      pages = {133--152},
      issn = {0025-5874},
      mrclass = {57S15 (53C20)},
      mrnumber = {3150196},
      mrreviewer = {Andrea Spiro},
      doi = {10.1007/s00209-013-1190-5},
      url = {https://doi.org/10.1007/s00209-013-1190-5},
      zblnumber = {1296.53066},
      }
  • [Go1] Go to document S. Goette, "Equivariant $\eta$-invariants on homogeneous spaces," Math. Z., vol. 232, iss. 1, pp. 1-42, 1999.
    @ARTICLE{Go1,
      author = {Goette, Sebastian},
      title = {Equivariant {$\eta$}-invariants on homogeneous spaces},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {232},
      year = {1999},
      number = {1},
      pages = {1--42},
      issn = {0025-5874},
      mrclass = {58J28 (58J20)},
      mrnumber = {1714278},
      mrreviewer = {Kai Köhler},
      doi = {10.1007/PL00004757},
      url = {https://doi.org/10.1007/PL00004757},
      zblnumber = {0941.58016},
      }
  • [Go] Go to document S. Goette, "Equivariant $\eta$-invariants and $\eta$-forms," J. Reine Angew. Math., vol. 526, pp. 181-236, 2000.
    @ARTICLE{Go,
      author = {Goette, Sebastian},
      title = {Equivariant {$\eta$}-invariants and {$\eta$}-forms},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {526},
      year = {2000},
      pages = {181--236},
      issn = {0075-4102},
      mrclass = {58J28 (58J20 58J35)},
      mrnumber = {1778304},
      mrreviewer = {Xiaonan Ma},
      doi = {10.1515/crll.2000.073},
      url = {https://doi.org/10.1515/crll.2000.073},
      zblnumber = {0974.58021},
      }
  • [Gohome] Go to document S. Goette, "Eta invariants of homogeneous spaces," Pure Appl. Math. Q., vol. 5, iss. 3, Special Issue: In honor of Friedrich Hirzebruch. Part 2, pp. 915-946, 2009.
    @ARTICLE{Gohome,
      author = {Goette, S.},
      title = {Eta invariants of homogeneous spaces},
      journal = {Pure Appl. Math. Q.},
      fjournal = {Pure and Applied Mathematics Quarterly},
      volume = {5},
      year = {2009},
      number = {3, Special Issue: In honor of Friedrich Hirzebruch. Part 2},
      pages = {915--946},
      issn = {1558-8599},
      mrclass = {58J28 (53C30)},
      mrnumber = {2532710},
      mrreviewer = {Thomas Schick},
      doi = {10.4310/PAMQ.2009.v5.n3.a2},
      url = {https://doi.org/10.4310/PAMQ.2009.v5.n3.a2},
      zblnumber = {1185.58010},
      }
  • [Gojems] Go to document S. Goette, "Adiabatic limits of Seifert fibrations, Dedekind sums, and the diffeomorphism type of certain 7-manifolds," J. Eur. Math. Soc. (JEMS), vol. 16, iss. 12, pp. 2499-2555, 2014.
    @ARTICLE{Gojems,
      author = {Goette, Sebastian},
      title = {Adiabatic limits of {S}eifert fibrations, {D}edekind sums, and the diffeomorphism type of certain 7-manifolds},
      journal = {J. Eur. Math. Soc. (JEMS)},
      fjournal = {Journal of the European Mathematical Society (JEMS)},
      volume = {16},
      year = {2014},
      number = {12},
      pages = {2499--2555},
      issn = {1435-9855},
      mrclass = {58J28 (53C21 57R20 57R55)},
      mrnumber = {3293803},
      mrreviewer = {Andrew Swann},
      doi = {10.4171/JEMS/492},
      url = {https://doi.org/10.4171/JEMS/492},
      zblnumber = {1311.58012},
      }
  • [GKS] Go to document S. Goette, N. Kitchloo, and K. Shankar, "Diffeomorphism type of the Berger space ${ SO}(5)/{ SO}(3)$," Amer. J. Math., vol. 126, iss. 2, pp. 395-416, 2004.
    @ARTICLE{GKS,
      author = {Goette, Sebastian and Kitchloo, Nitu and Shankar, Krishnan},
      title = {Diffeomorphism type of the {B}erger space {${\rm SO}(5)/{\rm SO}(3)$}},
      journal = {Amer. J. Math.},
      fjournal = {American Journal of Mathematics},
      volume = {126},
      year = {2004},
      number = {2},
      pages = {395--416},
      issn = {0002-9327},
      mrclass = {53C30 (58J28)},
      mrnumber = {2045506},
      mrreviewer = {Thomas Schick},
      doi = {10.1353/ajm.2004.0014},
      url = {https://doi.org/10.1353/ajm.2004.0014},
      zblnumber = {1066.53069},
      }
  • [GM] Go to document D. Gromoll and W. Meyer, "An exotic sphere with nonnegative sectional curvature," Ann. of Math. (2), vol. 100, pp. 401-406, 1974.
    @ARTICLE{GM,
      author = {Gromoll, Detlef and Meyer, Wolfgang},
      title = {An exotic sphere with nonnegative sectional curvature},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {100},
      year = {1974},
      pages = {401--406},
      issn = {0003-486X},
      mrclass = {53C20 (57D15)},
      mrnumber = {0375151},
      mrreviewer = {T. Hangan},
      doi = {10.2307/1971078},
      url = {https://doi.org/10.2307/1971078},
      zblnumber = {0293.53015},
      }
  • [GVZ] Go to document K. Grove, L. Verdiani, and W. Ziller, "An exotic $T_1\Bbb S^4$ with positive curvature," Geom. Funct. Anal., vol. 21, iss. 3, pp. 499-524, 2011.
    @ARTICLE{GVZ,
      author = {Grove, Karsten and Verdiani, Luigi and Ziller, Wolfgang},
      title = {An exotic {$T_1\Bbb S^4$} with positive curvature},
      journal = {Geom. Funct. Anal.},
      fjournal = {Geometric and Functional Analysis},
      volume = {21},
      year = {2011},
      number = {3},
      pages = {499--524},
      issn = {1016-443X},
      mrclass = {53C20 (53C21 53C25 57R55)},
      mrnumber = {2810857},
      mrreviewer = {David J. Wraith},
      doi = {10.1007/s00039-011-0117-8},
      url = {https://doi.org/10.1007/s00039-011-0117-8},
      zblnumber = {1230.53032},
      }
  • [GWZ] Go to document K. Grove, B. Wilking, and W. Ziller, "Positively curved cohomogeneity one manifolds and 3-Sasakian geometry," J. Differential Geom., vol. 78, iss. 1, pp. 33-111, 2008.
    @ARTICLE{GWZ,
      author = {Grove, Karsten and Wilking, Burkhard and Ziller, Wolfgang},
      title = {Positively curved cohomogeneity one manifolds and 3-{S}asakian geometry},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {78},
      year = {2008},
      number = {1},
      pages = {33--111},
      issn = {0022-040X},
      mrclass = {53C21 (53C26 57S25)},
      mrnumber = {2406265},
      mrreviewer = {Andrew Swann},
      doi = {10.4310/jdg/1197320603},
      url = {https://doi.org/10.4310/jdg/1197320603},
      zblnumber = {1145.53023},
      }
  • [GZ] Go to document K. Grove and W. Ziller, "Curvature and symmetry of Milnor spheres," Ann. of Math. (2), vol. 152, iss. 1, pp. 331-367, 2000.
    @ARTICLE{GZ,
      author = {Grove, Karsten and Ziller, Wolfgang},
      title = {Curvature and symmetry of {M}ilnor spheres},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {152},
      year = {2000},
      number = {1},
      pages = {331--367},
      issn = {0003-486X},
      mrclass = {53C20 (53C21 57R60 57S15)},
      mrnumber = {1792298},
      mrreviewer = {David J. Wraith},
      doi = {10.2307/2661385},
      url = {https://doi.org/10.2307/2661385},
      zblnumber = {0991.53016},
      }
  • [HR] Go to document A. J. Hanson and H. Römer, "Gravitational instanton contribution to Spin 3/2 axial anomaly," Phys. Lett. B, vol. 80, pp. 58-60, 1978.
    @ARTICLE{HR,
      author = {Hanson, A. J. and Römer, H.},
      title = {Gravitational instanton contribution to {S}pin~3/2 axial anomaly},
      journal = {Phys. Lett. B},
      volume = {80},
      year = {1978},
      pages = {58--60},
      doi = {10.1016/0370-2693(78)90306-4},
      url = {https://doi.org/10.1016/0370-2693(78)90306-4},
      zblnumber = {},
      }
  • [Hermann] Go to document R. Hermann, "A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle," Proc. Amer. Math. Soc., vol. 11, pp. 236-242, 1960.
    @ARTICLE{Hermann,
      author = {Hermann, Robert},
      title = {A sufficient condition that a mapping of {R}iemannian manifolds be a fibre bundle},
      journal = {Proc. Amer. Math. Soc.},
      fjournal = {Proceedings of the Amer. Math. Soc.},
      volume = {11},
      year = {1960},
      pages = {236--242},
      issn = {0002-9939},
      mrclass = {57.00 (53.00)},
      mrnumber = {0112151},
      mrreviewer = {Shoshichi Kobayashi},
      doi = {10.2307/2032963},
      url = {https://doi.org/10.2307/2032963},
      zblnumber = {0112.13701},
      }
  • [Hi] F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer-Verlag, Berlin, 1978, vol. 131.
    @BOOK{Hi,
      author = {Hirzebruch, Friedrich},
      title = {Topological Methods in Algebraic Geometry},
      series = {Classics Math.},
      note = {Second, Corrected Printing of the Third Edition},
      fjournal = {Grundlehren der Mathematischen Wissenschaften},
      journal = {Grundlehren Math. Wiss.},
      publisher = {Springer-Verlag, Berlin},
      volume = {131},
      year = {1978},
      pages = {234 pp.},
      isbn = {3-540-58663-6},
      mrclass = {57-02 (01A75 14-02)},
      mrnumber = {},
      zblnumber = {0376.14001},
      }
  • [KZ] Go to document V. Kapovitch and W. Ziller, "Biquotients with singly generated rational cohomology," Geom. Dedicata, vol. 104, pp. 149-160, 2004.
    @ARTICLE{KZ,
      author = {Kapovitch, Vitali and Ziller, Wolfgang},
      title = {Biquotients with singly generated rational cohomology},
      journal = {Geom. Dedicata},
      fjournal = {Geometriae Dedicata},
      volume = {104},
      year = {2004},
      pages = {149--160},
      issn = {0046-5755},
      mrclass = {22E46},
      mrnumber = {2043959},
      mrreviewer = {Samuel Evens},
      doi = {10.1023/B:GEOM.0000022860.89824.2f},
      url = {https://doi.org/10.1023/B:GEOM.0000022860.89824.2f},
      zblnumber = {1063.53055},
      }
  • [Kawa] Go to document T. Kawasaki, "The index of elliptic operators over $V$-manifolds," Nagoya Math. J., vol. 84, pp. 135-157, 1981.
    @ARTICLE{Kawa,
      author = {Kawasaki, Tetsuro},
      title = {The index of elliptic operators over {$V$}-manifolds},
      journal = {Nagoya Math. J.},
      fjournal = {Nagoya Mathematical Journal},
      volume = {84},
      year = {1981},
      pages = {135--157},
      issn = {0027-7630},
      mrclass = {58G10},
      mrnumber = {0641150},
      mrreviewer = {Th. Friedrich},
      doi = {10.1017/S0027763000019589},
      url = {https://doi.org/10.1017/S0027763000019589},
      zblnumber = {0437.58020},
      }
  • [KM] 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{KM,
      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 = {https://doi.org/10.2307/1970128},
      zblnumber = {0115.40505},
      }
  • [KiSh] Go to document N. Kitchloo and K. Shankar, "On complexes equivalent to $S^3$-bundles over $S^4$," Internat. Math. Res. Notices, iss. 8, pp. 381-394, 2001.
    @ARTICLE{KiSh,
      author = {Kitchloo, Nitu and Shankar, Krishnan},
      title = {On complexes equivalent to {$S^3$}-bundles over {$S^4$}},
      journal = {Internat. Math. Res. Notices},
      fjournal = {International Mathematics Research Notices},
      year = {2001},
      number = {8},
      pages = {381--394},
      issn = {1073-7928},
      mrclass = {55R25 (55P10)},
      mrnumber = {1827083},
      mrreviewer = {Donald M. Davis},
      doi = {10.1155/S1073792801000186},
      url = {https://doi.org/10.1155/S1073792801000186},
      zblnumber = {0981.55007},
      }
  • [KS] Go to document M. Kreck and S. Stolz, "A diffeomorphism classification of $7$-dimensional homogeneous Einstein manifolds with ${ SU}(3)\times{ SU}(2)\times{ U}(1)$-symmetry," Ann. of Math. (2), vol. 127, iss. 2, pp. 373-388, 1988.
    @ARTICLE{KS,
      author = {Kreck, Matthias and Stolz, Stephan},
      title = {A diffeomorphism classification of {$7$}-dimensional homogeneous {E}instein manifolds with {${\rm SU}(3)\times{\rm SU}(2)\times{\rm U}(1)$}-symmetry},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {127},
      year = {1988},
      number = {2},
      pages = {373--388},
      issn = {0003-486X},
      mrclass = {57R55 (53C25)},
      mrnumber = {0932303},
      mrreviewer = {Italo José Dejter},
      doi = {10.2307/2007059},
      url = {https://doi.org/10.2307/2007059},
      zblnumber = {0649.53029},
      }
  • [LM] B. H. Lawson Jr. and M. Michelsohn, Spin Geometry, Princeton Univ. Press, Princeton, NJ, 1989, vol. 38.
    @BOOK{LM,
      author = {Lawson, Jr., H. Blaine and Michelsohn, Marie-Louise},
      title = {Spin Geometry},
      series = {Princeton Math. Ser.},
      volume = {38},
      publisher = {Princeton Univ. Press, Princeton, NJ},
      year = {1989},
      pages = {xii+427},
      isbn = {0-691-08542-0},
      mrclass = {53-02 (53A50 53C20 57R75 58G10)},
      mrnumber = {1031992},
      mrreviewer = {N. J. Hitchin},
      zblnumber = {0688.57001},
      }
  • [Ma] Go to document X. Ma, "Functoriality of real analytic torsion forms," Israel J. Math., vol. 131, pp. 1-50, 2002.
    @ARTICLE{Ma,
      author = {Ma, Xiaonan},
      title = {Functoriality of real analytic torsion forms},
      journal = {Israel J. Math.},
      fjournal = {Israel Journal of Mathematics},
      volume = {131},
      year = {2002},
      pages = {1--50},
      issn = {0021-2172},
      mrclass = {58J52},
      mrnumber = {1942300},
      mrreviewer = {Thomas Schick},
      doi = {10.1007/BF02785849},
      url = {https://doi.org/10.1007/BF02785849},
      zblnumber = {1042.58019},
      }
  • [MM] Go to document R. R. Mazzeo and R. B. Melrose, "The adiabatic limit, Hodge cohomology and Leray’s spectral sequence for a fibration," J. Differential Geom., vol. 31, iss. 1, pp. 185-213, 1990.
    @ARTICLE{MM,
      author = {Mazzeo, Rafe R. and Melrose, Richard B.},
      title = {The adiabatic limit, {H}odge cohomology and {L}eray's spectral sequence for a fibration},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {31},
      year = {1990},
      number = {1},
      pages = {185--213},
      issn = {0022-040X},
      mrclass = {58A14 (55R20 58G15)},
      mrnumber = {1030670},
      mrreviewer = {Steven Rosenberg},
      doi = {10.4310/jdg/1214444094},
      url = {https://doi.org/10.4310/jdg/1214444094},
      zblnumber = {0702.58007},
      }
  • [Mi1] Go to document J. Milnor, "On manifolds homeomorphic to the $7$-sphere," Ann. of Math. (2), vol. 64, pp. 399-405, 1956.
    @ARTICLE{Mi1,
      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 = {https://doi.org/10.2307/1969983},
      zblnumber = {0072.18402},
      }
  • [Mi2] J. Milnor, Differentiable manifolds which are homotopy spheres, 1959.
    @MISC{Mi2,
      author = {Milnor, John},
      title = {Differentiable manifolds which are homotopy spheres},
      note = {mimeographed notes, Princeton Univ., Princeton, NJ},
      year = {1959},
      zblnumber = {0106.37001},
      }
  • [Mi3] J. Milnor, Collected Papers of John Milnor. III. Differential Topology, Amer. Math. Soc., Providence, RI, 2007.
    @BOOK{Mi3,
      author = {Milnor, John},
      title = {Collected Papers of {J}ohn {M}ilnor. {III}. Differential Topology},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2007},
      pages = {xvi+343},
      isbn = {978-0-8218-4230-0; 0-8218-4230-7},
      mrclass = {01A75 (01-06 55-03 55Q45 57-03 57Rxx)},
      mrnumber = {2307957},
      mrreviewer = {Serge L. Tabachnikov},
      zblnumber = {1122.01020},
      }
  • [PW] P. Petersen and F. Wilhelm, An exotic sphere with positive sectional curvature, 2008.
    @MISC{PW,
      author = {Petersen, P. and Wilhelm, F.},
      title = {An exotic sphere with positive sectional curvature},
      year = {2008},
      arxiv = {0805.0812},
      zblnumber = {},
      }
  • [Rochon] Go to document F. Rochon, "Pseudodifferential operators on manifolds with foliated boundaries," J. Funct. Anal., vol. 262, iss. 3, pp. 1309-1362, 2012.
    @ARTICLE{Rochon,
      author = {Rochon, Frédéric},
      title = {Pseudodifferential operators on manifolds with foliated boundaries},
      journal = {J. Funct. Anal.},
      fjournal = {Journal of Functional Analysis},
      volume = {262},
      year = {2012},
      number = {3},
      pages = {1309--1362},
      issn = {0022-1236},
      mrclass = {58J40 (35S05)},
      mrnumber = {2863864},
      mrreviewer = {Sandro Coriasco},
      doi = {10.1016/j.jfa.2011.11.007},
      url = {https://doi.org/10.1016/j.jfa.2011.11.007},
      zblnumber = {1238.58018},
      }
  • [SW] Go to document C. Searle and F. Wilhelm, "How to lift positive Ricci curvature," Geom. Topol., vol. 19, iss. 3, pp. 1409-1475, 2015.
    @ARTICLE{SW,
      author = {Searle, Catherine and Wilhelm, Frederick},
      title = {How to lift positive {R}icci curvature},
      journal = {Geom. Topol.},
      fjournal = {Geometry \& Topology},
      volume = {19},
      year = {2015},
      number = {3},
      pages = {1409--1475},
      issn = {1465-3060},
      mrclass = {53C20},
      mrnumber = {3352240},
      mrreviewer = {David J. Wraith},
      doi = {10.2140/gt.2015.19.1409},
      url = {https://doi.org/10.2140/gt.2015.19.1409},
      zblnumber = {1318.53029},
      }
  • [Sm1] Go to document S. Smale, "Generalized PoincarĂ©’s conjecture in dimensions greater than four," Ann. of Math. (2), vol. 74, pp. 391-406, 1961.
    @ARTICLE{Sm1,
      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 = {Morris W. Hirsch},
      doi = {10.2307/1970239},
      url = {https://doi.org/10.2307/1970239},
      zblnumber = {0099.39202},
      }
  • [Sm2] Go to document S. Smale, "On the structure of $5$-manifolds," Ann. of Math. (2), vol. 75, pp. 38-46, 1962.
    @ARTICLE{Sm2,
      author = {Smale, Stephen},
      title = {On the structure of {$5$}-manifolds},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {75},
      year = {1962},
      pages = {38--46},
      issn = {0003-486X},
      mrclass = {57.10},
      mrnumber = {0141133},
      mrreviewer = {A. Haefliger},
      doi = {10.2307/1970417},
      url = {https://doi.org/10.2307/1970417},
      zblnumber = {0101.16103},
      }
  • [Thurston] Go to document W. P. Thurston, The geometry and topology of three-manifolds.
    @MISC{Thurston,
      author = {Thurston, W. P.},
      title = {The geometry and topology of three-manifolds},
      note = {Electronic version~1.1, 2002},
      url = {http://www.msri.org/publications/books/gt3m/},
      zblnumber = {},
      }
  • [To] Go to document B. Totaro, "Cheeger manifolds and the classification of biquotients," J. Differential Geom., vol. 61, iss. 3, pp. 397-451, 2002.
    @ARTICLE{To,
      author = {Totaro, Burt},
      title = {Cheeger manifolds and the classification of biquotients},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {61},
      year = {2002},
      number = {3},
      pages = {397--451},
      issn = {0022-040X},
      mrclass = {53C35 (53C21 55P62 57R60)},
      mrnumber = {1979366},
      mrreviewer = {Vitali Kapovitch},
      doi ={10.4310/jdg/1090351529},
      url = {https://doi.org/10.4310/jdg/1090351529},
      zblnumber = {1071.53529},
      }
  • [VW] Go to document L. Verdiani and W. Ziller, "Concavity and rigidity in non-negative curvature," J. Differential Geom., vol. 97, iss. 2, pp. 349-375, 2014.
    @ARTICLE{VW,
      author = {Verdiani, Luigi and Ziller, Wolfgang},
      title = {Concavity and rigidity in non-negative curvature},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {97},
      year = {2014},
      number = {2},
      pages = {349--375},
      issn = {0022-040X},
      mrclass = {53C24 (53C21)},
      mrnumber = {3263509},
      mrreviewer = {Isabel M. C. Salavessa},
      doi = {10.4310/jdg/1405447808},
      url = {https://doi.org/10.4310/jdg/1405447808},
      url = {http://projecteuclid.org/euclid.jdg/1405447808},
      zblnumber = {1300.53039},
      }
  • [Wa1] Go to document C. T. C. Wall, "Classification of $(n-1)$-connected $2n$-manifolds," Ann. of Math. (2), vol. 75, pp. 163-189, 1962.
    @ARTICLE{Wa1,
      author = {Wall, C. T. C.},
      title = {Classification of {$(n-1)$}-connected {$2n$}-manifolds},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {75},
      year = {1962},
      pages = {163--189},
      issn = {0003-486X},
      mrclass = {57.10},
      mrnumber = {0145540},
      mrreviewer = {M. A. Kervaire},
      doi = {10.2307/1970425},
      url = {https://doi.org/10.2307/1970425},
      zblnumber = {0218.57022},
      }
  • [Wa2] Go to document C. T. C. Wall, "Classification problems in differential topology. VI. Classification of $(s-1)$-connected $(2s+1)$-manifolds," Topology, vol. 6, pp. 273-296, 1967.
    @ARTICLE{Wa2,
      author = {Wall, C. T. C.},
      title = {Classification problems in differential topology. {VI}. {C}lassification of {$(s-1)$}-connected {$(2s+1)$}-manifolds},
      journal = {Topology},
      fjournal = {Topology. An International Journal of Mathematics},
      volume = {6},
      year = {1967},
      pages = {273--296},
      issn = {0040-9383},
      mrclass = {57.10},
      mrnumber = {0216510},
      mrreviewer = {N. Kuiper},
      doi = {10.1016/0040-9383(67)90020-1},
      url = {https://doi.org/10.1016/0040-9383(67)90020-1},
      zblnumber = {0173.26102},
      }
  • [FW1] Go to document F. Wilhelm, "Exotic spheres with lots of positive curvatures," J. Geom. Anal., vol. 11, iss. 1, pp. 161-186, 2001.
    @ARTICLE{FW1,
      author = {Wilhelm, Frederick},
      title = {Exotic spheres with lots of positive curvatures},
      journal = {J. Geom. Anal.},
      fjournal = {The Journal of Geometric Analysis},
      volume = {11},
      year = {2001},
      number = {1},
      pages = {161--186},
      issn = {1050-6926},
      mrclass = {53C21 (53C20 57R55)},
      mrnumber = {1829354},
      mrreviewer = {Yaroslav V. Bazaĭkin},
      doi = {10.1007/BF02921960},
      url = {https://doi.org/10.1007/BF02921960},
      zblnumber = {1023.53024},
      }
  • [FW] Go to document F. Wilhelm, "An exotic sphere with positive curvature almost everywhere," J. Geom. Anal., vol. 11, iss. 3, pp. 519-560, 2001.
    @ARTICLE{FW,
      author = {Wilhelm, Frederick},
      title = {An exotic sphere with positive curvature almost everywhere},
      journal = {J. Geom. Anal.},
      fjournal = {The Journal of Geometric Analysis},
      volume = {11},
      year = {2001},
      number = {3},
      pages = {519--560},
      issn = {1050-6926},
      mrclass = {53C21 (53C20 57R55)},
      mrnumber = {1857856},
      mrreviewer = {David J. Wraith},
      doi = {10.1007/BF02922018},
      url = {https://doi.org/10.1007/BF02922018},
      zblnumber = {1039.53037},
      }
  • [DWi] Go to document D. L. Wilkens, "Closed $(s-1)$-connected $(2s+1)$-manifolds, $s=3,\,7$," Bull. London Math. Soc., vol. 4, pp. 27-31, 1972.
    @ARTICLE{DWi,
      author = {Wilkens, David L.},
      title = {Closed {$(s-1)$}-connected {$(2s+1)$}-manifolds, {$s=3,\,7$}},
      journal = {Bull. London Math. Soc.},
      fjournal = {The Bulletin of the London Mathematical Society},
      volume = {4},
      year = {1972},
      pages = {27--31},
      issn = {0024-6093},
      mrclass = {57D55},
      mrnumber = {0307258},
      mrreviewer = {C. Henry Edwards},
      doi = {10.1112/blms/4.1.27},
      url = {https://doi.org/10.1112/blms/4.1.27},
      zblnumber = {0241.57018},
      }
  • [Wr] Go to document D. Wraith, "Exotic spheres with positive Ricci curvature," J. Differential Geom., vol. 45, iss. 3, pp. 638-649, 1997.
    @ARTICLE{Wr,
      author = {Wraith, David},
      title = {Exotic spheres with positive {R}icci curvature},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {45},
      year = {1997},
      number = {3},
      pages = {638--649},
      issn = {0022-040X},
      mrclass = {53C21 (57R60)},
      mrnumber = {1472892},
      mrreviewer = {Ian Hambleton},
      doi = {10.4310/jdg/1214459846},
      url = {https://doi.org/10.4310/jdg/1214459846},
      zblnumber = {0910.53027},
      }

Authors

Sebastian Goette

Mathematisches Institut, Universität Freiburg, Germany

Martin Kerin

School of Mathematics, Statistics and Applied Mathematics, NUI Galway, Ireland

Krishnan Shankar

Department of Mathematics, The University of Oklahoma, Norman, OK, USA