Advertisement

The Homotopy Type of the Loops on \((n-1)\)-Connected \((2n+1)\)-Manifolds

  • Samik BasuEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

For \(n\ge 2\), we compute the homotopy groups of \((n-1)\)-connected closed manifolds of dimension \((2n+1)\). Away from the finite set of primes dividing the order of the torsion subgroup in homology, the p-local homotopy groups of M are determined by the rank of the free Abelian part of the homology. Moreover, we show that these p-local homotopy groups can be expressed as a direct sum of p-local homotopy groups of spheres. The integral homotopy type of the loop space is also computed and shown to depend only on the rank of the free Abelian part and the torsion subgroup.

Keywords

Homotopy groups Koszul duality Loop space Moore conjecture Quadratic algebra 

1991 Mathematics Subject Classification

Primary: 55P35 55Q52 Secondary: 16S37 57N15 

Notes

Acknowledgements

The author would like to thank the referee for pointing out the reference [7], and also for pointing out the history of the problem of loop space decompositions of highly connected manifolds.

References

  1. 1.
    J.F. Adams, On the non-existence of elements of Hopf invariant one. Ann. Math. 72(2), 20–104 (1960)MathSciNetCrossRefGoogle Scholar
  2. 2.
    J.F. Adams, Algebraic Topology—A Student’s Guide. London Mathematical Society Lecture Note Series, vol. 4 (Cambridge University Press, London, 1972)Google Scholar
  3. 3.
    S. Basu, S. Basu, Homotopy groups of certain highly connected manifolds via loop space homology. to appear in Osaka J. MathGoogle Scholar
  4. 4.
    S. Basu, S. Basu, Homotopy groups and periodic geodesics of closed 4-manifolds. Int. J. Math. 26, 1550059(2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    S. Basu, S. Basu, Homotopy groups of highly connected manifolds. Adv. Math. 337, 363–416 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    P. Beben, S. Theriault, The loop space homotopy type of simply-connected four-manifolds and their generalizations. Adv. Math. 262, 213–238 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    P. Beben, J. Wu, The homotopy type of a Poincaré duality complex after looping. Proc. Edinb. Math. Soc. 58(2), 581–616 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Berglund, K. Börjeson, Free loop space homology of highly connected manifolds. Forum Math. 29, 201–228 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    G.M. Bergman, The diamond lemma for ring theory. Adv. Math. 29, 178–218 (1978)MathSciNetCrossRefGoogle Scholar
  10. 10.
    A.L. Blakers, W.S. Massey, The homotopy groups of a triad. III. Ann. Math. 58(2), 409–417 (1953)MathSciNetCrossRefGoogle Scholar
  11. 11.
    P. Cartier, Remarques sur le théorème de Birkhoff–Witt. Ann. Scuola Norm. Sup. Pisa 12(3), 1–4 (1958)Google Scholar
  12. 12.
    F.R. Cohen, J.C. Moore, J.A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres. Ann. Math. 110(2), 549–565 (1979)MathSciNetCrossRefGoogle Scholar
  13. 13.
    F.R. Cohen, J.C. Moore, J.A. Neisendorfer, Torsion in homotopy groups. Ann. Math. 109(2), 121–168 (1979)MathSciNetCrossRefGoogle Scholar
  14. 14.
    P.M. Cohn, A remark on the Birkhoff-Witt theorem. J. Lond. Math. Soc. 38, 197–203 (1963)MathSciNetCrossRefGoogle Scholar
  15. 15.
    H. Duan, C. Liang, Circle bundles over 4-manifolds. Arch. Math. (Basel) 85, 278–282 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Y. Félix, S. Halperin, J.-C. Thomas, Rational Homotopy Theory. Graduate Texts in Mathematics, vol. 205 (Springer, New York, 2001)CrossRefGoogle Scholar
  17. 17.
    B. Gray, On the sphere of origin of infinite families in the homotopy groups of spheres. Topology 8, 219–232 (1969)MathSciNetCrossRefGoogle Scholar
  18. 18.
    P.J. Hilton, On the homotopy groups of the union of spheres. J. Lond. Math. Soc. 30, 154–172 (1955)MathSciNetCrossRefGoogle Scholar
  19. 19.
    I.M. James, On the suspension sequence. Ann. Math. 65(2), 74–107 (1957)MathSciNetCrossRefGoogle Scholar
  20. 20.
    P. Lalonde, A. Ram, Standard Lyndon bases of Lie algebras and enveloping algebras. Trans. Am. Math. Soc. 347, 1821–1830 (1995)MathSciNetCrossRefGoogle Scholar
  21. 21.
    M. Lazard, Sur les algèbres enveloppantes universelles de certaines algèbres de Lie. Publ. Sci. Univ. Alger. Sér. A. 1(1954), 281–294 (1955)zbMATHGoogle Scholar
  22. 22.
    J.-L. Loday, B. Vallette, Algebraic Operads. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346 (Springer, Heidelberg, 2012)CrossRefGoogle Scholar
  23. 23.
    M. Lothaire, Combinatorics on Words. Cambridge Mathematical Library (Cambridge University Press, Cambridge, 1997). With a foreword by Roger Lyndon and a preface by Dominique Perrin, Corrected reprint of the 1983 original, with a new preface by PerrinGoogle Scholar
  24. 24.
    C.A. McGibbon, J.A. Neisendorfer, Various applications of Haynes Miller’s theorem, in Conference on Algebraic Topology in Honor of Peter Hilton (Saint John’s, Nfld., 1983). Contemporary Mathematics, vol. 37 (American Mathematical Society, Providence, 1985), pp. 91–98Google Scholar
  25. 25.
    H. Miller, The Sullivan conjecture on maps from classifying spaces. Ann. Math. 120(2), 39–87 (1984)MathSciNetCrossRefGoogle Scholar
  26. 26.
    T.J. Miller, On the formality of \((k-1)\)- connected compact manifolds of dimension less than or equal to \(4k-2\). Ill. J. Math. 23, 253–258 (1979)MathSciNetzbMATHGoogle Scholar
  27. 27.
    J. Milnor, D. Husemoller, Symmetric Bilinear Forms (Springer, New York, 1973). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73Google Scholar
  28. 28.
    J.C. Moore, The double suspension and \(p\)-primary components of the homotopy groups of spheres. Bol. Soc. Mat. Mex. 1(2), 28–37 (1956)Google Scholar
  29. 29.
    J.A. Neisendorfer, \(3\)- primary exponents. Math. Proc. Camb. Philos. Soc. 90, 63–83 (1981)MathSciNetCrossRefGoogle Scholar
  30. 30.
    J.A. Neisendorfer, P.S. Selick, Some examples of spaces with or without exponents, in Current Trends in Algebraic Topology, Part 1 (London, Ont., 1981). CMS Conference Proceedings, vol. 2 (American Mathematical Society, Providence, 1982), pp. 343–357Google Scholar
  31. 31.
    A. Polishchuk, L. Positselski, Quadratic Algebras. University Lecture Series, vol. 37 (American Mathematical Society, Providence, 2005)Google Scholar
  32. 32.
    D.C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres. Pure and Applied Mathematics, vol. 121 (Academic, Orlando, 1986)Google Scholar
  33. 33.
    H. Samelson, Classifying spaces and spectral sequences. Am. J. Math. 75, 744–752 (1953)CrossRefGoogle Scholar
  34. 34.
    P. Selick, Odd primary torsion in \(\pi _{k}(S^{3})\). Topology 17, 407–412 (1978)MathSciNetCrossRefGoogle Scholar
  35. 35.
    J.-P. Serre, Homologie singulière des espaces fibrés. Applications. Ann. Math. 54(2), 425–505 (1951)MathSciNetCrossRefGoogle Scholar
  36. 36.
    S. Smale, On the structure of \(5\)-manifolds. Ann. Math. 75(2), 38–46 (1962)Google Scholar
  37. 37.
    H. Toda, On the double suspension \(E^2\). J. Inst. Polytech. Osaka City Univ. Ser. A. 7, 103–145 (1956)Google Scholar
  38. 38.
    H. Toda, Composition Methods in Homotopy Groups of Spheres. Annals of Mathematics Studies, vol. 49 (Princeton University Press, Princeton, 1962)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Stat-Math Unit, Indian Statistical InstituteKolkataIndia

Personalised recommendations