Exponential growth and an asymptotic formula for the ranks of homotopy groups of a finite 1-connected complex

Abstract

Let $X$ be an $n$-dimensional, finite, simply connected CW complex and set $\alpha_X =\limsup_i (\log\text{ rank}\, \pi_i(X))/i$. We prove that either $\text{rank}\, \pi_i(X) = 0\,, i\geq 2n\,,$ or else that $0\lt \alpha_X\lt \infty$ and that for any $\varepsilon>0$ there is a $K=K(\varepsilon )$ such that \[e^{(\alpha_X -\varepsilon)k}\leq \sum_{i=k+2}^{k+n} \text{rank}\, \pi_i(X) \, \leq e^{(\alpha_X + \varepsilon)k}\,, \quad \mbox{for all } k\geq K\,. \] In particular, this sum grows exponentially in $k$.

  • [AH] Go to document J. F. Adams and P. J. Hilton, "On the chain algebra of a loop space," Comment. Math. Helv., vol. 30, pp. 305-330, 1956.
    @article {AH, MRKEY = {MR0077929},
      AUTHOR = {Adams, J. F. and Hilton, P. J.},
      TITLE = {On the chain algebra of a loop space},
      JOURNAL = {Comment. Math. Helv.},
      FJOURNAL = {Commentarii Mathematici Helvetici},
      VOLUME = {30},
      YEAR = {1956},
      PAGES = {305--330},
      ISSN = {0010-2571},
      MRCLASS = {55.0X},
      MRNUMBER = {17,1119b},
      MRREVIEWER = {W. S. Massey},
      DOI = {10.1007/BF02564350},
      }
  • [Av] L. L. Avramov, "Local algebra and rational homotopy," in Algebraic Homotopy and Local Algebra, Paris: Soc. Math. France, 1984, vol. 113, pp. 15-43.
    @incollection {Av, MRKEY = {MR749041},
      AUTHOR = {Avramov, Luchezar L.},
      TITLE = {Local algebra and rational homotopy},
      BOOKTITLE = {Algebraic Homotopy and Local Algebra},
      VENUE = {{L}uminy, 1982},
      SERIES = {Astérisque},
      VOLUME = {113},
      PAGES = {15--43},
      PUBLISHER = {Soc. Math. France},
      ADDRESS = {Paris},
      YEAR = {1984},
      MRCLASS = {55P62 (13D03 13D25)},
      MRNUMBER = {85j:55021},
      MRREVIEWER = {Yves F{é}lix},
      }
  • [FH] Go to document Y. Félix and S. Halperin, "Rational LS category and its applications," Trans. Amer. Math. Soc., vol. 273, iss. 1, pp. 1-38, 1982.
    @article {FH, MRKEY = {MR664027},
      AUTHOR = {F{é}lix, Yves and Halperin, Stephen},
      TITLE = {Rational {LS} category and its applications},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {273},
      YEAR = {1982},
      NUMBER = {1},
      PAGES = {1--38},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {55P62 (55M30 55P50)},
      MRNUMBER = {84h:55011},
      MRREVIEWER = {Jean-Michel Lemaire},
      DOI = {10.2307/1999190},
      }
  • [rad] Go to document Y. Félix, S. Halperin, C. Jacobsson, C. Löfwall, and J. Thomas, "The radical of the homotopy Lie algebra," Amer. J. Math., vol. 110, iss. 2, pp. 301-322, 1988.
    @article {rad, MRKEY = {MR935009},
      AUTHOR = {F{é}lix, Yves and Halperin, Stephen and Jacobsson, Carl and L{ö}fwall, Clas and Thomas, Jean-Claude},
      TITLE = {The radical of the homotopy {L}ie algebra},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {110},
      YEAR = {1988},
      NUMBER = {2},
      PAGES = {301--322},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {55P62 (13H99 17B56 55P50)},
      MRNUMBER = {89d:55029},
      MRREVIEWER = {Lex Renner},
      DOI = {10.2307/2374504},
      }
  • [IHES] Go to document Y. Félix, S. Halperin, and J. Thomas, "The homotopy Lie algebra for finite complexes," Inst. Hautes Études Sci. Publ. Math., iss. 56, pp. 179-202 (1983), 1982.
    @article {IHES, MRKEY = {MR686046},
      AUTHOR = {F{é}lix, Yves and Halperin, Stephen and Thomas, Jean-Claude},
      TITLE = {The homotopy {L}ie algebra for finite complexes},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {56},
      YEAR = {1982},
      PAGES = {179--202 (1983)},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {55P62 (13H15)},
      MRNUMBER = {85c:55010},
      MRREVIEWER = {Jean-Michel Lemaire},
      URL = {http://www.numdam.org/item?id=PMIHES_1982__56__179_0},
      }
  • [Lie] Go to document Y. Félix, S. Halperin, and J. Thomas, "Lie algebras of polynomial growth," J. London Math. Soc., vol. 43, iss. 3, pp. 556-566, 1991.
    @article {Lie, MRKEY = {MR1113393},
      AUTHOR = {F{é}lix, Yves and Halperin, Stephen and Thomas, Jean-Claude},
      TITLE = {Lie algebras of polynomial growth},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {43},
      YEAR = {1991},
      NUMBER = {3},
      PAGES = {556--566},
      ISSN = {0024-6107},
      CODEN = {JLMSAK},
      MRCLASS = {57T05 (16P90 17B30 17B70)},
      MRNUMBER = {92i:57034},
      MRREVIEWER = {Marc Aubry},
      DOI = {10.1112/jlms/s2-43.3.556},
      }
  • [FHT] Y. Félix, S. Halperin, and J. Thomas, Rational Homotopy Theory, New York: Springer-Verlag, 2001.
    @book {FHT,
      author = {F{é}lix, Yves and Halperin, Stephen and Thomas, Jean-Claude},
      TITLE = {Rational Homotopy Theory},
      SERIES = {Grad. Texts in Math.},
      NUMBER = {205},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2001},
      PAGES = {xxxiv+535},
      ISBN = {0-387-95068-0},
      MRCLASS = {55P62 (18Gxx 55U35)},
      MRNUMBER = {2002d:55014},
      MRREVIEWER = {John F. Oprea},
      ZBLNUMBER = {0961.55002},
      INVZBLNUMBER = {0939.55010},
      }
  • [F-H] Go to document J. B. Friedlander and S. Halperin, "An arithmetic characterization of the rational homotopy groups of certain spaces," Invent. Math., vol. 53, iss. 2, pp. 117-133, 1979.
    @article {F-H, MRKEY = {MR560410},
      AUTHOR = {Friedlander, John B. and Halperin, Stephen},
      TITLE = {An arithmetic characterization of the rational homotopy groups of certain spaces},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {53},
      YEAR = {1979},
      NUMBER = {2},
      PAGES = {117--133},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {55P62},
      MRNUMBER = {81f:55006b},
      MRREVIEWER = {K.-T. Chen},
      DOI = {10.1016/0022-314X(79)90005-2},
      }
  • [Roos] Y. Félix, S. Halperin, and J. Thomas, "Growth and Lie brackets in the homotopy Lie algebra," Homology Homotopy Appl., vol. 4, iss. 2, part 1, pp. 219-225 (electronic), 2002.
    @article {Roos, MRKEY = {MR1918190},
      AUTHOR = {F{é}lix, Yves and Halperin, Stephen and Thomas, Jean-Claude},
      TITLE = {Growth and {L}ie brackets in the homotopy {L}ie algebra},
      NOTE = {The Roos Festschrift volume, 1},
      JOURNAL = {Homology Homotopy Appl.},
      FJOURNAL = {Homology, Homotopy and Applications},
      VOLUME = {4},
      YEAR = {2002},
      NUMBER = {2, part 1},
      PAGES = {219--225 (electronic)},
      ISSN = {1512-0139},
      MRCLASS = {55P62 (17B70 55P35 55Q15)},
      MRNUMBER = {2003g:55014},
      MRREVIEWER = {Gregory Lupton},
      }
  • [AF] Go to document Y. Félix, S. Halperin, and J. Thomas, "An asymptotic formula for the ranks of homotopy groups," Topology Appl., vol. 153, iss. 18, pp. 3430-3436, 2006.
    @article {AF, MRKEY = {MR2270596},
      AUTHOR = {F{é}lix, Yves and Halperin, Stephen and Thomas, Jean-Claude},
      TITLE = {An asymptotic formula for the ranks of homotopy groups},
      JOURNAL = {Topology Appl.},
      FJOURNAL = {Topology and its Applications},
      VOLUME = {153},
      YEAR = {2006},
      NUMBER = {18},
      PAGES = {3430--3436},
      ISSN = {0166-8641},
      CODEN = {TIAPD9},
      MRCLASS = {55P62 (17B70 55P35)},
      MRNUMBER = {2007i:55010},
      MRREVIEWER = {Antonio G{ó}mez-Tato},
      DOI = {10.1016/j.topol.2006.03.004},
      }
  • [exp3] Go to document Y. Félix, S. Halperin, and J. Thomas, "Exponential growth of Lie algebras of finite global dimension," Proc. Amer. Math. Soc., vol. 135, iss. 5, pp. 1575-1578 (electronic), 2007.
    @article {exp3, MRKEY = {MR2276669},
      AUTHOR = {F{é}lix, Yves and Halperin, Stephen and Thomas, Jean-Claude},
      TITLE = {Exponential growth of {L}ie algebras of finite global dimension},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {135},
      YEAR = {2007},
      NUMBER = {5},
      PAGES = {1575--1578 (electronic)},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {17B70 (55P35 55P62)},
      MRNUMBER = {2008b:17048},
      MRREVIEWER = {Gregory Lupton},
      DOI = {10.1090/S0002-9939-07-08721-7},
      }
  • [ranks] Go to document Y. Félix, S. Halperin, and J. Thomas, "The ranks of the homotopy groups of a space of finite LS category," Expo. Math., vol. 25, iss. 1, pp. 67-76, 2007.
    @article {ranks, MRKEY = {MR2286835},
      AUTHOR = {F{é}lix, Yves and Halperin, Stephen and Thomas, Jean-Claude},
      TITLE = {The ranks of the homotopy groups of a space of finite {LS} category},
      JOURNAL = {Expo. Math.},
      FJOURNAL = {Expositiones Mathematicae},
      VOLUME = {25},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {67--76},
      ISSN = {0723-0869},
      MRCLASS = {55P62 (17B70 55P35)},
      MRNUMBER = {2007k:55016},
      MRREVIEWER = {Kathryn P. Hess},
      DOI = {10.1016/j.exmath.2006.04.002},
      }
  • [Kos] Go to document J. Koszul, "Homologie et cohomologie des algèbres de Lie," Bull. Soc. Math. France, vol. 78, pp. 65-127, 1950.
    @article {Kos, MRKEY = {MR0036511},
      AUTHOR = {Koszul, Jean-Louis},
      TITLE = {Homologie et cohomologie des algèbres de {L}ie},
      JOURNAL = {Bull. Soc. Math. France},
      FJOURNAL = {Bulletin de la Société Mathématique de France},
      VOLUME = {78},
      YEAR = {1950},
      PAGES = {65--127},
      ISSN = {0037-9484},
      MRCLASS = {56.0X},
      MRNUMBER = {12,120g},
      MRREVIEWER = {C. Chevalley},
      URL = {http://www.numdam.org/item?id=BSMF_1950__78__65_0},
      }
  • [Lambrechts] Go to document P. Lambrechts, "Analytic properties of Poincaré series of spaces," Topology, vol. 37, iss. 6, pp. 1363-1370, 1998.
    @article {Lambrechts, MRKEY = {MR1632936},
      AUTHOR = {Lambrechts, Pascal},
      TITLE = {Analytic properties of {P}oincaré series of spaces},
      JOURNAL = {Topology},
      FJOURNAL = {Topology. An International Journal of Mathematics},
      VOLUME = {37},
      YEAR = {1998},
      NUMBER = {6},
      PAGES = {1363--1370},
      ISSN = {0040-9383},
      CODEN = {TPLGAF},
      MRCLASS = {55P35 (55P62)},
      MRNUMBER = {99e:55011},
      MRREVIEWER = {Yves F{é}lix},
      DOI = {10.1016/S0040-9383(97)00083-9},
      }
  • [May] Go to document J. P. May, "The cohomology of restricted Lie algebras and of Hopf algebras," J. Algebra, vol. 3, pp. 123-146, 1966.
    @article {May, MRKEY = {MR0193126},
      AUTHOR = {May, J. P.},
      TITLE = {The cohomology of restricted {L}ie algebras and of {H}opf algebras},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {3},
      YEAR = {1966},
      PAGES = {123--146},
      ISSN = {0021-8693},
      MRCLASS = {18.20 (17.30)},
      MRNUMBER = {33 \#1347},
      MRREVIEWER = {J. F. Adams},
      DOI = {10.1016/0021-8693(66)90009-3},
      }
  • [MM] Go to document J. W. Milnor and J. C. Moore, "On the structure of Hopf algebras," Ann. of Math., vol. 81, pp. 211-264, 1965.
    @article {MM, MRKEY = {MR0174052},
      AUTHOR = {Milnor, John W. and Moore, John C.},
      TITLE = {On the structure of {H}opf algebras},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {81},
      YEAR = {1965},
      PAGES = {211--264},
      ISSN = {0003-486X},
      MRCLASS = {55.40 (55.34)},
      MRNUMBER = {30 \#4259},
      MRREVIEWER = {I. M. James},
      DOI = {10.2307/1970615},
      }
  • [infinitesimal] Go to document D. Sullivan, "Infinitesimal computations in topology," Inst. Hautes Études Sci. Publ. Math., iss. 47, pp. 269-331 (1978), 1977.
    @article {infinitesimal, MRKEY = {MR0646078},
      AUTHOR = {Sullivan, Dennis},
      TITLE = {Infinitesimal computations in topology},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {47},
      YEAR = {1977},
      PAGES = {269--331 (1978)},
      ISSN = {0073-8301},
      MRCLASS = {57D99 (55D99 58A10)},
      MRNUMBER = {58 \#31119},
      MRREVIEWER = {J. F. Adams},
      URL = {http://www.numdam.org/item?id=PMIHES_1977__47__269_0},
      }
  • [VW] J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge: Cambridge University Press, 1992.
    @book {VW, MRKEY = {MR1207813},
      AUTHOR = {van Lint, J. H. and Wilson, R. M.},
      TITLE = {A Course in Combinatorics},
      PUBLISHER = {Cambridge University Press},
      ADDRESS = {Cambridge},
      YEAR = {1992},
      PAGES = {xii+530},
      ISBN = {0-521-41057-6; 0-521-42260-4},
      MRCLASS = {05-01 (05Axx 05Exx 90B10 94B25)},
      MRNUMBER = {94g:05003},
      MRREVIEWER = {George E. Andrews},
      }

Authors

Yves Felix

Département de Mathématique
Université catholique de Louvain
Bât. M. de Hemptinne
Chemin du Cyclotron, 2
1348 Louvain-la-Neuve
Belgium

Steve Halperin

College of Computer, Mathematical and Physical Sciences
AV Williams Building
University of Maryland
College Park, MD 20742
United States

Jean-Claude Thomas

Université d’Angers and CNRS
2 Bd Lavoisier
49045 Angers Cedex
France