Abstract
We define partial quasi-morphisms on the group of Hamiltonian diffeomorphisms of the cotangent bundle using the spectral invariants in Lagrangian Floer homology with conormal boundary conditions, where the product compatible with the PSS isomorphism and the homological intersection product is lacking.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
More generally, they defined the mapping \(\mu _a\) for each \(a\in H^1(M)\), and proved that it has the properties analogous to those of a partial quasi-morphism (due to the later results of Shelukhin [24] and Kislev–Shelukhin[12], \(\mu _a\) give rise to genuine quasi-morphisms for some M). If \(M=\mathbb {T}^n\), \(\mu _p(\phi _1^H)=\overline{H}(p)\), where \(\overline{H}\) is the Viterbo’s homogenization [30] of H and \(p\in \mathbb {R}^n\cong H^1(\mathbb {T}^n)\) [15] (see also [14]).
References
Abbondandolo, A., Schwarz, M.: Floer homology of cotangent bundles and the loop product. Geom. Topol. 14(3), 1569–1722 (2010)
Auroux, D.: A Beginner’s Introduction to Fukaya categories. arXiv:1301.7056 (2013)
Banyaga, A.: Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique. Commentarii Mathematici Helvetici 53, 174–227 (1978)
Biran, P., Cornea, O.: Lagrangian quantum homology. arXiv:0808.3989
Duretić, J.: Piunikhin-Salamon-Schwarz isomorphisms and spectral invariants for conormal bundle. Publication de l’Institut Mathématique, tome 102(116), 17–47 (2017)
Entov, M.: Quasi-morphisms and quasi-states in symplectic topology. In: Proceedings of the International Congress of Mathematicians, Seoul, vol. II, pp. 1147–1171 (2014)
Entov, M., Polterovich, L.: Quasi-states and symplectic intersections. Comment. Math. Helv. 81(1), 75–99 (2006)
Humilière, V., Leclercq, R., Seyfaddini, S.: Reduction of symplectic homeomorphisms. Ann. Sci. Éc. Norm. Supér. (4) 49(3), 633–668 (2016)
Kasturirangan, R., Oh, Y.-G.: Floer homology of open subsets and a relative version of Arnold’s conjecture. Math. Z. 236, 151–189 (2001)
Katić, J., Milinković, D., Nikolić, J.: Spectral invariants in Lagrangian Floer homology of open subset. Differ. Geom. Appl. 53, 220–267 (2017)
Katić, J., Milinković, D., Nikolić, J.: Spectral numbers and manifolds with boundary. Topol. Methods Nonlinear Anal. 55(2), 617–653 (2020)
Kislev, A., Shelukhin, E.: Bounds on spectral norms and barcodes. Geom. Topol. 25, 3257–3350 (2021)
Leclercq, R., Zapolsky, F.: Spectral invariants for monotone Lagrangians. J. Topol. Anal. 10(03), 627–700 (2018)
Monzner, A., Zapolsky, F.: A comparison of symplectic homogenization and Calabi quasi-states. J. Topol. Anal. 3(3), 243–263 (2011)
Monzner, A., Vichery, N., Zapolsky, F.: Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization. J. Mod. Dyn. Issue 2, 205–249 (2012)
Oh, Y.-G.: Symplectic topology as the geometry of action functional, I. Relative Floer theory on the cotangent bundle. J. Differ. Geom. 46(3), 499–577 (1997)
Oh, Y.-G.: Symplectic topology as the geometry of action functional, II–pants product and cohomological invariants. Commun. Anal. Geom. 7(1), 1–55 (1999)
Oh, Y.-G.: Chain level Floer theory and Hofer’s geometry of the Hamiltonian diffeomorphism group. Asian J. Math. 6, 579–624 (2002)
Oh, Y.-G.: Construction of spectral invariants of Hamiltonian paths for closed symplectic manifolds. In: The Breadth of Symplectic and Poisson Geometry. Progress of Mathematics 232, pp. 525–570. Birkhäuser, Boston (2005)
Polterovich, L., Rosen, D.: Function Theory on Symplectic Manifolds, CRM Monograph Series, vol. 34. American Mathematical Society, Providence (2014)
Poźniak, M.: Floer homology, Novikov rings and clean intersections, Ph.D. thesis, University of Warwick (1994)
Robbin, J., Salamon, D.: The Maslov index for paths. Topology 32, 827–844 (1993)
Robbin, J., Salamon, D.: The spectral flow and the Maslov index. Bull. Lond. Math. Soc. 27, 1–33 (1995)
Shelukhin, E.: Viterbo conjecture for Zoll symmetric spaces, preprint arXiv:1811.05552 (2018)
Schwarz, M.: On the action spectrum for closed symplectically aspherical manifolds. Pac. J. Math. 193(2), 419–461 (2000)
Schwarz, M.: Cohomology Operations from \(S^1\)–Cobordisms in Floer Homology, Ph.D. thesis, ETH Zurich (1995)
Usher, M.: Spectral numbers in Floer theories. Compos. Math. 144(6), 1581–1592 (2008)
Usher, M.: Duality in filtered Floer-Novikov complexes. J. Topol Anal. 2(2), 233–258 (2010)
Viterbo, C.: Symplectic topology as the geometry of generating functions. Math. Ann. 292, 685–710 (1992)
Viterbo, C.: Symplectic homogenization, preprint arXiv:0801.0206 (2007)
Acknowledgements
The third author is grateful to Octav Cornea for useful discussion at the workshop Current trends in symplectic topology in Montreal 2019, and to Morimichi Kawasaki for the email discussion that helped us clarify some concepts related to this paper. The authors thank Vukašin Stojisavljević for useful discussions. We are also grateful to the anonymous referee for many valuable comments and suggestions, and especially for pointing out an error in the proof of conjugation invariance property in the first version of the manuscript. The authors’ research is partially supported by the contract of the Ministry of Education, Science and Technological Development of the Republic of Serbia no. 451-03-9/2021-14/200104.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Katić, J., Milinković, D. & Nikolić, J. A Note on Partial Quasi-Morphisms and Products in Lagrangian Floer Homology in Cotangent Bundles. Mediterr. J. Math. 19, 149 (2022). https://doi.org/10.1007/s00009-022-02043-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-022-02043-0