A bound on the cohomology of quasiregularly elliptic manifolds

Abstract

We show that a closed, connected and orientable Riemannian manifold $M$ of dimension $d$ that admits a nonconstant quasiregular mapping from $\mathbb {R}^d$ must have bounded dimension of the cohomology independent of the distortion of the map. The dimension of the degree $l$ de Rham cohomology of $M$ is bounded above by $\binom {d}{l}$. This is a sharp upper bound that proves the Bonk-Heinonen conjecture. A corollary of this theorem answers an open problem posed by Gromov in 1981. He asked whether there exists a \hbox $d$-dimensional, simply connected manifold that does not admit a quasiregular mapping from $\mathbb {R}^d$. Our result gives an affirmative answer to this question.

  • [bojarskiiwaniec] Go to document B. Bojarski and T. Iwaniec, "Analytical foundations of the theory of quasiconformal mappings in ${\bf R}^{n}$," Ann. Acad. Sci. Fenn. Ser. A I Math., vol. 8, iss. 2, pp. 257-324, 1983.
    @ARTICLE{bojarskiiwaniec,
      author = {Bojarski, B. and Iwaniec, T.},
      title = {Analytical foundations of the theory of quasiconformal mappings in {${\bf R}\sp{n}$}},
      journal = {Ann. Acad. Sci. Fenn. Ser. A I Math.},
      fjournal = {Annales Academiae Scientiarum Fennicae. Series A I. Mathematica},
      volume = {8},
      year = {1983},
      number = {2},
      pages = {257--324},
      issn = {0066-1953},
      mrclass = {30C60 (58E15)},
      mrnumber = {0731786},
      mrreviewer = {H. Renelt},
      doi = {10.5186/aasfm.1983.0806},
      url = {https://doi.org/10.5186/aasfm.1983.0806},
      zblnumber = {0548.30016},
      }
  • [bonkheinonen] Go to document M. Bonk and J. Heinonen, "Quasiregular mappings and cohomology," Acta Math., vol. 186, iss. 2, pp. 219-238, 2001.
    @ARTICLE{bonkheinonen,
      author = {Bonk, Mario and Heinonen, Juha},
      title = {Quasiregular mappings and cohomology},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {186},
      year = {2001},
      number = {2},
      pages = {219--238},
      issn = {0001-5962},
      mrclass = {30C65 (53C21 58A12)},
      mrnumber = {1846030},
      mrreviewer = {Bianca Stroffolini},
      doi = {10.1007/BF02401840},
      url = {https://doi.org/10.1007/BF02401840},
      zblnumber = {1088.30011},
      }
  • [bonkpoggi] M. Bonk and P. Poggi-Corradini, The Rickman-Picard theorem, 2019.
    @MISC{bonkpoggi,
      author = {Bonk, Mario and Poggi-Corradini, P.},
      title = {The {R}ickman-{P}icard theorem},
      note = {to appear in \emph{Ann. Acad. Sci. Fenn. Ser. A I Math.} {\bf 44},
      no.~2},
      year = {2019},
      arxiv = {1807.07683},
      }
  • [botttu] Go to document R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Springer-Verlag, New York-Berlin, 1982, vol. 82.
    @BOOK{botttu,
      author = {Bott, Raoul and Tu, Loring W.},
      title = {Differential Forms in Algebraic Topology},
      series = {Grad. Texts in Math.},
      volume = {82},
      publisher = {Springer-Verlag, New York-Berlin},
      year = {1982},
      pages = {xiv+331},
      isbn = {0-387-90613-4},
      mrclass = {57R19 (55-02 58-01 58A12)},
      mrnumber = {0658304},
      mrreviewer = {Hansklaus Rummler},
      zblnumber = {0496.55001},
      doi = {10.1007/978-1-4757-3951-0},
     }
  • [donaldsonsullivan] Go to document S. K. Donaldson and D. P. Sullivan, "Quasiconformal $4$-manifolds," Acta Math., vol. 163, iss. 3-4, pp. 181-252, 1989.
    @ARTICLE{donaldsonsullivan,
      author = {Donaldson, S. K. and Sullivan, D. P.},
      title = {Quasiconformal {$4$}-manifolds},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {163},
      year = {1989},
      number = {3-4},
      pages = {181--252},
      issn = {0001-5962},
      mrclass = {57N13 (53C20 58D29)},
      mrnumber = {1032074},
      mrreviewer = {L. Guillou},
      doi = {10.1007/BF02392736},
      url = {https://doi.org/10.1007/BF02392736},
      zblnumber = {0704.57008},
      }
  • [drasinpankka] Go to document D. Drasin and P. Pankka, "Sharpness of Rickman’s Picard theorem in all dimensions," Acta Math., vol. 214, iss. 2, pp. 209-306, 2015.
    @ARTICLE{drasinpankka,
      author = {Drasin, David and Pankka, Pekka},
      title = {Sharpness of {R}ickman's {P}icard theorem in all dimensions},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {214},
      year = {2015},
      number = {2},
      pages = {209--306},
      issn = {0001-5962},
      mrclass = {30C65},
      mrnumber = {3372169},
      mrreviewer = {David Matthew Freeman},
      doi = {10.1007/s11511-015-0125-x},
      url = {https://doi.org/10.1007/s11511-015-0125-x},
      zblnumber = {1326.30025},
      }
  • [eremenkolewis] Go to document A. Eremenko and J. L. Lewis, "Uniform limits of certain $A$-harmonic functions with applications to quasiregular mappings," Ann. Acad. Sci. Fenn. Ser. A I Math., vol. 16, iss. 2, pp. 361-375, 1991.
    @ARTICLE{eremenkolewis,
      author = {Eremenko, Alexandre and Lewis, John L.},
      title = {Uniform limits of certain {$A$}-harmonic functions with applications to quasiregular mappings},
      journal = {Ann. Acad. Sci. Fenn. Ser. A I Math.},
      fjournal = {Annales Academiae Scientiarum Fennicae. Series A I. Mathematica},
      volume = {16},
      year = {1991},
      number = {2},
      pages = {361--375},
      issn = {0066-1953},
      mrclass = {35J60 (30C65 35B40)},
      mrnumber = {1139803},
      mrreviewer = {Juha Heinonen},
      doi = {10.5186/aasfm.1991.1609},
      url = {https://doi.org/10.5186/aasfm.1991.1609},
      zblnumber = {0727.35022},
      }
  • [gehring] Go to document F. W. Gehring, "The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping," Acta Math., vol. 130, pp. 265-277, 1973.
    @ARTICLE{gehring,
      author = {Gehring, F. W.},
      title = {The {$L\sp{p}$}-integrability of the partial derivatives of a quasiconformal mapping},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {130},
      year = {1973},
      pages = {265--277},
      issn = {0001-5962},
      mrclass = {30A60},
      mrnumber = {0402038},
      mrreviewer = {P. Caraman},
      doi = {10.1007/BF02392268},
      url = {https://doi.org/10.1007/BF02392268},
      zblnumber = {0258.30021},
      }
  • [gromov1981] Go to document M. Gromov, "Hyperbolic manifolds, groups and actions," in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Princeton Univ. Press, Princeton, N.J., 1981, vol. 97, pp. 183-213.
    @INCOLLECTION{gromov1981,
      author = {Gromov, M.},
      title = {Hyperbolic manifolds, groups and actions},
      booktitle = {Riemann Surfaces and Related Topics: {P}roceedings of the 1978 {S}tony {B}rook {C}onference},
      venue = {State {U}niv. {N}ew {Y}ork, {S}tony {B}rook, {N}.{Y}., 1978},
      series = {Ann. of Math. Stud.},
      volume = {97},
      pages = {183--213},
      publisher = {Princeton Univ. Press, Princeton, N.J.},
      year = {1981},
      mrclass = {53C15 (53C45 58F17)},
      mrnumber = {0624814},
      mrreviewer = {M. Rees},
      zblnumber = {0467.53035},
      doi = {10.1515/9781400881550},
      }
  • [gromov2007] Go to document M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, English ed., Birkhäuser Boston, Inc., Boston, MA, 2007.
    @BOOK{gromov2007,
      author = {Gromov, Misha},
      title = {Metric Structures for {R}iemannian and Non-{R}iemannian Spaces},
      series = {Modern Birkhäuser Classics},
      edition = {{E}nglish},
      note = {based on the 1981 French original, with appendices by M. Katz, P. Pansu and S. Semmes, translated from the French by Sean Michael Bates},
      publisher = {Birkhäuser Boston, Inc., Boston, MA},
      year = {2007},
      pages = {xx+585},
      isbn = {978-0-8176-4582-3; 0-8176-4582-9},
      mrclass = {53C23 (53-02)},
      mrnumber = {2307192},
      zblnumber = {1113.53001},
      doi = {10.1007/978-0-8176-4583-0}
    }
  • [guilleminpollack] Go to document V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974.
    @BOOK{guilleminpollack,
      author = {Guillemin, Victor and Pollack, Alan},
      title = {Differential Topology},
      publisher = {Prentice-Hall, Inc., Englewood Cliffs, N.J.},
      year = {1974},
      pages = {xvi+222},
      mrclass = {58-01 (57-01)},
      mrnumber = {0348781},
      mrreviewer = {K. H. Mayer},
      zblnumber = {05788282},
      doi = {10.1090/chel/370},
      }
  • [iwanieclutoborski] Go to document T. Iwaniec and A. Lutoborski, "Integral estimates for null Lagrangians," Arch. Rational Mech. Anal., vol. 125, iss. 1, pp. 25-79, 1993.
    @ARTICLE{iwanieclutoborski,
      author = {Iwaniec, Tadeusz and Lutoborski, Adam},
      title = {Integral estimates for null {L}agrangians},
      journal = {Arch. Rational Mech. Anal.},
      fjournal = {Archive for Rational Mechanics and Analysis},
      volume = {125},
      year = {1993},
      number = {1},
      pages = {25--79},
      issn = {0003-9527},
      mrclass = {58E35 (46E35 49J10)},
      mrnumber = {1241286},
      doi = {10.1007/BF00411477},
      url = {https://doi.org/10.1007/BF00411477},
      zblnumber = {0793.58002},
      }
  • [iwaniecmartin] Go to document T. Iwaniec and G. Martin, "Quasiregular mappings in even dimensions," Acta Math., vol. 170, iss. 1, pp. 29-81, 1993.
    @ARTICLE{iwaniecmartin,
      author = {Iwaniec, Tadeusz and Martin, Gaven},
      title = {Quasiregular mappings in even dimensions},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {170},
      year = {1993},
      number = {1},
      pages = {29--81},
      issn = {0001-5962},
      mrclass = {30C65},
      mrnumber = {1208562},
      mrreviewer = {Matti Vuorinen},
      doi = {10.1007/BF02392454},
      url = {https://doi.org/10.1007/BF02392454},
      zblnumber = {0785.30008},
      }
  • [jormakka] J. Jormakka, "The existence of quasiregular mappings from ${\bf R}^3$ to closed orientable $3$-manifolds," Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, iss. 69, p. 44, 1988.
    @ARTICLE{jormakka,
      author = {Jormakka, Jorma},
      title = {The existence of quasiregular mappings from {${\bf R}^3$} to closed orientable {$3$}-manifolds},
      journal = {Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes},
      fjournal = {Annales Academiae Scientiarum Fennicae. Series A I. Mathematica Dissertationes},
      number = {69},
      year = {1988},
      pages = {44},
      issn = {0355-0087},
      mrclass = {57M99 (26B99 30C60)},
      mrnumber = {0973719},
      mrreviewer = {N. V. Ivanov},
      zblnumber = {0662.57007},
      }
  • [kangasniemi] I. Kangasniemi, Sharp cohomological bound For uniformly quasiregularly elliptic manifolds, 2017.
    @MISC{kangasniemi,
      author = {Kangasniemi, I.},
      title = {Sharp cohomological bound For uniformly quasiregularly elliptic manifolds},
      arxiv = {1711.11410},
      year = {2017},
      zblnumber = {},
      }
  • [lehtovirtanen] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Second ed., Springer-Verlag, New York-Heidelberg, 1973, vol. 126.
    @BOOK{lehtovirtanen,
      author = {Lehto, O. and Virtanen, K. I.},
      title = {Quasiconformal Mappings in the Plane},
      edition = {Second},
      note = {translated from the German by K. W. Lucas},
      series = {Grundlehren Math. Wissen.},
      volume = {126},
      publisher = {Springer-Verlag, New York-Heidelberg},
      year = {1973},
      pages = {viii+258},
      mrclass = {30A60},
      mrnumber = {0344463},
      zblnumber = {0267.30016},
      }
  • [lewis] Go to document J. L. Lewis, "Picard’s theorem and Rickman’s theorem by way of Harnack’s inequality," Proc. Amer. Math. Soc., vol. 122, iss. 1, pp. 199-206, 1994.
    @ARTICLE{lewis,
      author = {Lewis, John L.},
      title = {Picard's theorem and {R}ickman's theorem by way of {H}arnack's inequality},
      journal = {Proc. Amer. Math. Soc.},
      fjournal = {Proceedings of the American Mathematical Society},
      volume = {122},
      year = {1994},
      number = {1},
      pages = {199--206},
      issn = {0002-9939},
      mrclass = {30C65},
      mrnumber = {1195483},
      mrreviewer = {Ilkka Holopainen},
      doi = {10.2307/2160861},
      url = {https://doi.org/10.2307/2160861},
      zblnumber = {0807.30010},
      }
  • [mattilarickman] Go to document P. Mattila and S. Rickman, "Averages of the counting function of a quasiregular mapping," Acta Math., vol. 143, iss. 3-4, pp. 273-305, 1979.
    @ARTICLE{mattilarickman,
      author = {Mattila, P. and Rickman, S.},
      title = {Averages of the counting function of a quasiregular mapping},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {143},
      year = {1979},
      number = {3-4},
      pages = {273--305},
      issn = {0001-5962},
      mrclass = {30C60},
      mrnumber = {0549779},
      mrreviewer = {V. Mićić},
      doi = {10.1007/BF02392097},
      url = {https://doi.org/10.1007/BF02392097},
      zblnumber = {0443.30043},
      }
  • [pankka] Go to document P. Pankka, "Mappings of bounded mean distortion and cohomology," Geom. Funct. Anal., vol. 20, iss. 1, pp. 229-242, 2010.
    @ARTICLE{pankka,
      author = {Pankka, Pekka},
      title = {Mappings of bounded mean distortion and cohomology},
      journal = {Geom. Funct. Anal.},
      fjournal = {Geometric and Functional Analysis},
      volume = {20},
      year = {2010},
      number = {1},
      pages = {229--242},
      issn = {1016-443X},
      mrclass = {30C65 (53C23 58A12)},
      mrnumber = {2647140},
      mrreviewer = {Leonid V. Kovalev},
      doi = {10.1007/s00039-010-0054-y},
      url = {https://doi.org/10.1007/s00039-010-0054-y},
      zblnumber = {1203.30023},
      }
  • [rickman1980] Go to document S. Rickman, "On the number of omitted values of entire quasiregular mappings," J. Analyse Math., vol. 37, pp. 100-117, 1980.
    @ARTICLE{rickman1980,
      author = {Rickman, Seppo},
      title = {On the number of omitted values of entire quasiregular mappings},
      journal = {J. Analyse Math.},
      fjournal = {Journal d'Analyse Mathématique},
      volume = {37},
      year = {1980},
      pages = {100--117},
      issn = {0021-7670},
      mrclass = {30D60},
      mrnumber = {0583633},
      mrreviewer = {Tadashi Kuroda},
      doi = {10.1007/BF02797681},
      url = {https://doi.org/10.1007/BF02797681},
      zblnumber = {0451.30012},
      }
  • [rickman1985] Go to document S. Rickman, "The analogue of Picard’s theorem for quasiregular mappings in dimension three," Acta Math., vol. 154, iss. 3-4, pp. 195-242, 1985.
    @ARTICLE{rickman1985,
      author = {Rickman, Seppo},
      title = {The analogue of {P}icard's theorem for quasiregular mappings in dimension three},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {154},
      year = {1985},
      number = {3-4},
      pages = {195--242},
      issn = {0001-5962},
      mrclass = {30C60 (30D35)},
      mrnumber = {0781587},
      mrreviewer = {Gaven J. Martin},
      doi = {10.1007/BF02392472},
      url = {https://doi.org/10.1007/BF02392472},
      zblnumber = {0617.30024},
      }
  • [rickman1993] Go to document S. Rickman, Quasiregular Mappings, Springer-Verlag, Berlin, 1993, vol. 26.
    @BOOK{rickman1993,
      author = {Rickman, Seppo},
      title = {Quasiregular Mappings},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {26},
      publisher = {Springer-Verlag, Berlin},
      year = {1993},
      pages = {x+213},
      isbn = {3-540-56648-1},
      mrclass = {30C65},
      mrnumber = {1238941},
      mrreviewer = {Gaven J. Martin},
      doi = {10.1007/978-3-642-78201-5},
      url = {https://doi.org/10.1007/978-3-642-78201-5},
      zblnumber = {0816.30017},
      }
  • [rickman2006] S. Rickman, "Simply connected quasiregularly elliptic 4-manifolds," Ann. Acad. Sci. Fenn. Math., vol. 31, iss. 1, pp. 97-110, 2006.
    @ARTICLE{rickman2006,
      author = {Rickman, Seppo},
      title = {Simply connected quasiregularly elliptic 4-manifolds},
      journal = {Ann. Acad. Sci. Fenn. Math.},
      fjournal = {Annales Academiæ Scientiarum Fennicæ . Mathematica},
      volume = {31},
      year = {2006},
      number = {1},
      pages = {97--110},
      issn = {1239-629X},
      mrclass = {30F20 (30C65 57M12 57N13)},
      mrnumber = {2210111},
      mrreviewer = {Alessia Cattabriga},
      zblnumber = {1116.30011},
      }
  • [vsc] Go to document T. N. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1992, vol. 100.
    @BOOK{vsc,
      author = {Varopoulos, N. Th. and Saloff-Coste, L. and Coulhon, T.},
      title = {Analysis and Geometry on Groups},
      series = {Cambridge Tracts in Math.},
      volume = {100},
      publisher = {Cambridge University Press, Cambridge},
      year = {1992},
      pages = {xii+156},
      isbn = {0-521-35382-3},
      mrclass = {43A80 (47D03 47F05 58G11 60B15)},
      mrnumber = {1218884},
      mrreviewer = {A. Hulanicki},
      zblnumber = {0813.22003},
      doi = {10.1017/CBO9780511662485},
      }

Authors

Eden Prywes

University of California, Los Angeles, CA