Skip to main content

Topological coHochschild homology and the homology of free loop spaces

Mathematische Zeitschrift volume 301, pages 411–454 (2022)Cite this article

Abstract

We study the homology of free loop spaces via techniques arising from the theory of topological coHochschild homology (coTHH). Topological coHochschild homology is a topological analogue of the classical theory of coHochschild homology for coalgebras. We produce new spectrum-level structure on coTHH of suspension spectra as well as new algebraic structure in the coBökstedt spectral sequence for computing coTHH. These new techniques allow us to compute the homology of free loop spaces in several new cases, extending known calculations.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Aguiar, M., Mahajan, S.: Monoidal Functors, Species and Hopf Algebras, Volume 29 of CRM Monograph Series. American Mathematical Society, Providence (2010) (With forewords by Kenneth Brown and Stephen Chase and André Joyal)

  2. Angeltveit, V., Rognes, J.: Hopf algebra structure on topological Hochschild homology. Algebr. Geom. Topol. 5, 1223–1290 (2005)

    Article  MathSciNet  Google Scholar 

  3. Bayındır, H., Péroux, M.: Spanier–Whitehead duality for topological coHochschild homology (2020). arXiv:math.AT/2012.03966

  4. Bohmann, A.M., Gerhardt, T., Høgenhaven, A., Shipley, B., Ziegenhagen, S.: Computational tools for topological coHochschild homology. Topol. Appl. 235, 185–213 (2018)

    Article  MathSciNet  Google Scholar 

  5. Bökstedt, M.: The topological Hochschild homology of \(\mathbb{Z}\) and \(\mathbb{Z}/p\)(unpublished)

  6. Bousfield, A.K., Kan, D.M.: Homotopy Limits, Completions and Localizations. Lecture Notes in Mathematics, vol. 304. Springer, Berlin (1972)

    Book  Google Scholar 

  7. Bousfield, A.K., Kan, D.M.: Pairings and products in the homotopy spectral sequence. Trans. Am. Math. Soc. 177, 319–343 (1973)

    Article  MathSciNet  Google Scholar 

  8. Bousfield, A.K., Kan, D.M.: A second quadrant homotopy spectral sequence. Trans. Am. Math. Soc. 177, 305–318 (1973)

    Article  MathSciNet  Google Scholar 

  9. Brzezinski, T., Wisbauer, R.: Corings and Comodules. London Mathematical Society Lecture Note Series, vol. 309. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  10. Burghelea, D., Fiedorowicz, Z.: Cyclic homology and algebraic \(K\)-theory of spaces. II. Topology 25(3), 303–317 (1986)

    Article  MathSciNet  Google Scholar 

  11. Chas, M., Sullivan, D.: String topology (1999). arXiv:math.GT/9911159

  12. Cohen, R.L., Jones, J.D.S.: A homotopy theoretic realization of string topology. Math. Ann. 324(4), 773–798 (2002)

    Article  MathSciNet  Google Scholar 

  13. 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), Volume 215 of Progress in Mathematics, pp. 77–92. Birkhäuser, Basel (2004)

  14. Doi, Y.: Homological coalgebra. J. Math. Soc. Jpn. 33(1), 31–50 (1981)

    Article  MathSciNet  Google Scholar 

  15. Eilenberg, S., Moore, J.C.: Homology and fibrations. I. Coalgebras, cotensor product and its derived functors. Comment. Math. Helv. 40, 199–236 (1966)

    Article  MathSciNet  Google Scholar 

  16. Elmendorf, A.D., Kriz, I., Mandell, M.A., May, J.P.: Rings, Modules, and Algebras in Stable Homotopy theory, volume 47 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1997) (with an appendix by M. Cole)

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

    Article  MathSciNet  Google Scholar 

  18. Gromoll, D., Meyer, W.: Periodic geodesics on compact riemannian manifolds. J. Differ. Geom. 3, 493–510 (1969)

    Article  MathSciNet  Google Scholar 

  19. Hess, K., Parent, P.-E., Scott, J.: CoHochschild homology of chain coalgebras. J. Pure Appl. Algebra 213(4), 536–556 (2009)

    Article  MathSciNet  Google Scholar 

  20. Hess, Kathryn, Shipley, Brooke: Invariance properties of coHochschild homology. J. Pure Appl. Algebra 225(2), 106505 (2021)

    Article  MathSciNet  Google Scholar 

  21. Joyal, A.: Quasi-categories and Kan complexes. J. Pure Appl. Algebra 175(1–3), 207–222 (2002)

    Article  MathSciNet  Google Scholar 

  22. Klanderman, S.: Computations of relative topological coHochschild homology. arXiv:2108.07863 (2021)

  23. Kuribayashi, K.: The Hochschild cohomology ring of the singular Cochain algebra of a space. Ann. Inst. Fourier (Grenoble) 61(5):1779–1805 (2012) (2011)

  24. Kuribayashi, K., Menichi, L., Naito, T.: Behavior of the Eilenberg–Moore spectral sequence in derived string topology. Topol. Appl. 164, 24–44 (2014)

    Article  MathSciNet  Google Scholar 

  25. Kuribayashi, K., Yamaguchi, T.: The cohomology algebra of certain free loop spaces. Fund. Math. 154(1), 57–73 (1997)

    Article  MathSciNet  Google Scholar 

  26. Lurie, J.: Elliptic cohomology I. September 2016 version. Available on author’s homepage

  27. Lurie, J.: Higher algebra. September 2017 version. Available on author’s homepage

  28. Malkiewich, C.: Cyclotomic structure in the topological Hochschild homology of \(DX\). Algebra Geom. Topol. 17(4), 2307–2356 (2017)

    Article  MathSciNet  Google Scholar 

  29. McClure, J., Schwänzl, R., Vogt, R.: \(T\!H\!H(R)\cong R\otimes S^1\) for \(E_\infty \) ring spectra. J. Pure Appl. Algebra 121(2), 137–159 (1997)

  30. Menichi, L.: The cohomology ring of free loop spaces. Homol. Homotopy Appl. 3(1), 193–224 (2001)

    Article  MathSciNet  Google Scholar 

  31. Péroux, M.: Rigidification of connective comodules (2020). arXiv:2006.09398

  32. Riehl, E., Verity, D.: Elements of \(\infty \)-category theory. Version of May 2021. Available on the first author’s homepage

  33. Smith, L.: The Eilenberg–Moore spectral sequence and the mod \(2\) cohomology of certain free loop spaces. Ill. J. Math. 28(3), 516–522 (1984)

    MathSciNet  MATH  Google Scholar 

  34. Moss, E.: Sweedler. Hopf algebras. Mathematics Lecture Note Series. W. A. Benjamin Inc, New York (1969)

    Google Scholar 

  35. Weibel, C.A.: An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)

    Book  Google Scholar 

  36. Ziller, W.: The free loop space of globally symmetric spaces. Invent. Math. 41(1), 1–22 (1977)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors express their gratitude to the organizers of the Women in Topology II Workshop and the Banff International Research Station, where this collaboration began. We thank Vigleik Angeltveit, Ben Antieau, David Chan, Paul Goerss, Kathryn Hess, Sarah Klanderman, Maximilien Péroux, Emily Riehl, and Stephanie Ziegenhagen for helpful conversations related to this work. The authors also thank an anonymous referee for helpful comments. This research was supported by the National Science Foundation [DMS-1710534 and DMS-2104300 to Bohmann, DMS-1810575 to Gerhardt, and DMS-1811278 to Shipley]. Some of this work was done while the second author was in residence at the Mathematical Sciences Research Institute in Berkeley, CA (supported by the National Science Foundation under Grant DMS-1440140) during the Spring 2020 semester. The first and third authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the program ‘Homotopy harnessing higher structures’ during which some of the work on this paper was carried out. Work during this program was supported by EPSRC Grant no EP/K032208/1.

Author information

Authors and Affiliations

  1. Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN, 37240, USA

    Anna Marie Bohmann

  2. Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI, 48824, USA

    Teena Gerhardt

  3. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 508 SEO m/c 249, 851 S. Morgan Street, Chicago, IL, 60607-7045, USA

    Brooke Shipley

Authors
  1. Anna Marie Bohmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Teena Gerhardt
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Brooke Shipley
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anna Marie Bohmann.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bohmann, A.M., Gerhardt, T. & Shipley, B. Topological coHochschild homology and the homology of free loop spaces. Math. Z. 301, 411–454 (2022). https://doi.org/10.1007/s00209-021-02879-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-021-02879-4

Keywords

  • Topological Hochschild homology
  • Coalgebra
  • Free loop spaces

Mathematics Subject Classification

  • Primary: 55P35
  • 16T15
  • Secondary: 55P43
  • 13D03
  • 16T05
  • 55T99