Skip to main content
SpringerLink
Log in

On the free loop spaces of a toric space

  • Original Article
  • Published:
Boletín de la Sociedad Matemática Mexicana Aims and scope Submit manuscript

Abstract

In this note, it is shown that the Hilbert–Poincaré series for the rational homology of the free loop space on a moment-angle complex is a rational function if and only if the moment-angle complex is a product of odd spheres and a disk. A partial result is included for the Davis–Januszkiewicz spaces. The opportunity is taken to correct the result (Bahri et al., Proceedings of the Steklov Institute of Mathematics, Russian Academy of Sciences, vol. 286, pp. 219–223. doi:10.1134/S0081543814060121, 2014) which used a theorem from Berglund and Jöllenbeck (J Algebra 315:249–273, 2007).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bahri, A., Bendersky, M., Cohen, F.R., Gitler, S.: The polyhedral product functor: a method of computation for moment–angle complexes, arrangements and related spaces. Adv. Math. 225, 1634–1668 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bahri, A., Bendersky, M., Cohen, F.R., Gitler, S.: On the rational type of moment–angle complexes. In: Proceedings of the Steklov Institute of Mathematics, Russian Academy of Sciences, vol. 286, pp. 219–223 (2014). doi:10.1134/S0081543814060121

  3. Berglund, A., Jöllenbeck, M.: On the Golod property of Stanly–Reisner rings. J. Algebra 315, 249–273 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buchstaber, V., Panov, T.: Torus actions, combinatorial topology and homological algebra. Uspekhi Mat. Nauk 55(5, 3) (106) (2000) (Russian). [Russ. Math. Surv. 55(5), 825–921 (2000) (English translation)]. arXiv:math.AT/0010073

  5. Buchstaber, V., Panov, T.: Torus actions and their applications in topology and combinatorics. In: AMS University Lecture Series, vol. 24 (2002)

  6. Buchstaber, V., Panov, T.: Toric topology. In: Mathematical Surveys and Monographs, vol. 204. American Mathematical Society, Providence, RI (2015). arXiv:1210.2368

  7. Cohen, R.L., Jones, J.D.S., Yan, J.: The loop homology algebra of spheres and projective spaces. In: Categorical Decomposition Techniques in Algebraic Topology, Isle of Skye, 2001. Progr. Math., vol. 215, pp, 77–92. Birkhauser, Basel (2004)

  8. Denham, G., Suciu, A.: Moment–angle complexes, monomial ideals and Massey products. Pure Appl. Math. Q 3(1), 25–60 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fadell, E., Husseini, S.: Infinite cup length in free loop spaces with an application to a problem of the \(N\)-body type. Ann. Inst. H. Poincare Anal. Non Lineaire 9(3), 305–319 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Félix, Y., Halperin, S., Thomas, J.-C.: On the growth of the homology of a free loop space (English summary). Pure Appl. Math. Q. 9(1), 167–187 (2013)

  11. Felix, Y., Halperin, S.: Formal spaces with finite-dimensional rational homotopy. Trans. Am. Math. Soc. 270(2), 575–588 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  12. Felix, Y., Halperin, S., Thomas, J.-C.: Rational homotopy theory. In: Graduate Texts in Mathematics. Springer, New York (2002)

  13. Goodwillie, T.G.: Cyclic homology, derivations, and the free loopspace. Topology 24(2), 187–215 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gurvich, M.: Some results on the topology of quasitoric manifolds and their equivariant mapping spaces. Ph.D. thesis, Series: UC San Diego Electronic Theses and Dissertations (2008)

  15. Lambrechts, P.: On the Betti numbers of the free loop space of a coformal space. J. Pure Appl. Algebra 161(1–2), 177–192 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. McCullough, J., Peeva, I.: Infinite graded free resolutions. In: Eisenbud, Iyengar, Singh, Stafford, Van den Bergh (eds.) Commutative Algebra and Noncommutative Algebraic Geometry. Math. Sci. Res. Inst. Publ., Cambridge University Press (2016)

  17. Neisendorfer, J.: Algebraic Methods in Unstable Homotopy Theory. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  18. Neisendorfer, J., Miller, T.: Formal and coformal spaces. Ill. J. Math. 22(2), 565–580 (1978)

  19. Roos, J.-E.: Homology of free loop spaces, cyclic homology and nonrational Poincaré-Betti series in commutative algebra. In: Algebra Some Current Trends (Varna, 1986). Lecture Notes in Math., vol. 1352, pp. 173–189. Springer, Berlin (1988)

  20. Seeliger, N.: Loop homology algebra of spheres and complex projective spaces. Forum Math. 26(4), 967–981 (2011). doi:10.1515/FORM.2011.161. (ISSN (Online): 1435-5337, ISSN (Print): 0933-7741)

  21. Serre, J.-P.: Un exemple de série de Poincaré non rationnelle, (French) Nederl. Akad. Wetensch. Indag. Math. 41(4), 469–471 (1979)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to Ran Levi and Kathryn Hess for useful suggestions and also to Jason McCullough. The comments of the referee have improved the exposition. A. Bahri was supported in part by Grant No. 210386 from the Simons Foundation.

Author information

Authors and Affiliations

  1. Department of Mathematics, Rider University, Lawrenceville, NJ, 08648, USA

    A. Bahri

  2. Department of Mathematics, CUNY, East 695 Park Avenue, New York, NY, 10065, USA

    M. Bendersky

  3. Department of Mathematics, University of Rochester, Rochester, NY, 14625, USA

    F. R. Cohen

  4. Department of Mathematics, Cinvestav, San Pedro Zacatenco, Apartado Postal 14-740, 07360, Mexico, DF, Mexico

    S. Gitler

Authors
  1. A. Bahri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Bendersky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. R. Cohen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. Gitler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Bahri.

Additional information

This paper is dedicated to Samuel Gitler Hammer who brought us much joy and interest in mathematics.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bahri, A., Bendersky, M., Cohen, F.R. et al. On the free loop spaces of a toric space. Bol. Soc. Mat. Mex. 23, 257–265 (2017). https://doi.org/10.1007/s40590-016-0124-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40590-016-0124-8

Keywords

Mathematics Subject Classification

Search

Navigation