Spectral numbers and manifolds with boundary
Keywords
Spectral numbers, Floer homology, Lagrangian submanifolds, manifolds with boundaryAbstract
We consider a smooth submanifold $N$ with a smooth boundary in an ambient closed manifold $M$ and assign a spectral invariant $c(\alpha,H)$ to every singular homological class $\alpha\in H_*(N)$ and a Hamiltonian $H$ defined on the cotangent bundle $T^*M$. We also derive certain properties of spectral numbers, for example we prove that spectral invariants $c_\pm(H,N)$ associated to the whole Floer homology $HF_*(H,N:M)$ of the submanifold $N$, are the limits of decreasing nested family of open sets.References
A. Abbondadolo and M. Schwarz, Notes on Floer homology and loop space homology, Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, NATO Sci. Ser. II Math. Phys. Chem., vol. 217, Springer, Dordrecht (2006), 74–108.
P. Albers, A Lagrangian Piunikhin–Salamon–Schwarz morphism and two comparison homomorphisms in Floer homology, Int. Math. Res. Not. IMRN 2008 (2008), no. 4.
D. Auroux, A Beginner’s Introduction to Fukaya Categories, Contact and Symplectic Topology. Bolyai Society Mathematical Studies (F. Bourgeois, V. Colin V. and A. Stipsicz, eds), vol. 26, Springer, Cham, 2014.
Y. Chekanov, Hofer’s symplectic energy and Lagrangian intersections, Publ. Newton Inst., vol. 8, Cambridge University Press, 1996, pp. 296–306.
J. Duretić, Piunikhin–Salamon–Schwarz isomorphisms and spectral invariants for conormal bundle, Publication de l’Institut Mathématique, t. 102 (116), (2017), pp. 17–47.
Y. Eliashberg and L. Polterovich, Symplectic quasi-states on the quadric surface and Lagrangian submanifolds, (2010), ArXiv:1006.2501v1.
U. Frauenfelder and F. Schlenk, Hamiltonian dynamics on convex symplectic manifolds, Israel J. Math. 159 (2007), 1–56.
V. Humilière, R. Leclercq and S. Seyfaddini, Coisotropic rigidity and C 0 −symplectic geometry, Duke Math. J. 164 (2015), no. 4, 767–799.
A.A. Kosinski, Differential Manifolds, Academic Press, San Diego, 1993.
R. Kasturirangan and Y.-G. Oh, Floer homology for open subsets and a relative version of Arnold’s conjecture, Math. Z. 236 (2001), 151–189.
J. Katić and D. Milinković, Piunikhin–Salamon–Schwarz isomorphism for Lagrangian intersections, Differential Geom. Appl. 22 (2005), 215–227.
J. Katić, D. Milinković and J. Nikolić, Spectral invariants in Lagrangian Floer homology of open subset, Differential Geom. Appl. 53 (2017), 220–267.
J. Katić, D. Milinković and T. Simčević, Isomorphism between Morse nad Lagrangian Floer cohomology rings, Rocky Mountain J. Math. 41 (2011), no. 3, 789–811.
S. Lanzat, Hamiltonian Floer homology for compact convex symplectic manifolds, Beitr. Algebra Geom. 57 (2016), 361–390.
R. Leclercq, Spectral invariants in Lagrangian Floer theory, J. Modern Dynamics 2 (2008), 249–286.
D. Milinković, Morse homology for generating functions of Lagrangian submanifolds, Trans. Amer. Math. Soc. 351 (1999), no. 10, 3953–3974.
D. Milinković, On equivalence of two constructions of invariants of Lagrangian submanifolds, Pacific J. Math. 195 (2000), no. 2, 371–415.
D. Milinković, Geodesics on the space of Lagrangian submanifolds in cotangent bundles, Proc. Amer. Math. Soc. 129 (2001), 1843–1851.
D. Milinković, Action spectrum and Hofer’s distance between Lagrangian submanifolds, Differential Geom. Appl. 17 (2002), 69–81.
A. Monzner, N. Vichery and F. Zapolsky, Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization, J. Mod. Dyn. 2 (2012), 205–249.
D. Nadler and E. Zaslow, Constructible sheaves and the Fukaya category. J. Amer. Math. Soc. 22 (2009), 233–286.
Y.-G. Oh, Symplectic topology as the geometry of action functional I, J. Differential Geom. 46 (1997), 499–577.
Y.-G. Oh, Symplectic topology as the geometry of action functional II – pants product and cohomological invariants, Comm. Anal. Geom. 7 (1999), 1–55.
Y.-G. Oh, Naturality of Floer homology of open subsets in Lagrangian intersection theory, Proc. of Pacific Rim Geometry Conference 1996, International Press (1998), 261–280.
Y.-G. Oh, Geometry of generating functions and Lagrangian spectral invariants (2013), arXiv:1206.4788 (2013).
L. Polterovich and D. Rosen, Function theory on symplectic manifolds, CRM Monograph Series, vol. 34, 2014.
S. Piunikhin, D. Salamon and M. Schwarz, Symplectic Floer–Donaldson theory and quantum cohomology, Contact and Symplectic Geometry, Publ. Newton Instit., vol. 8, 1996, Cambridge Univ. Press, Cambridge, pp. 171–200.
M. Schwarz, Morse Homology, Progress in Math., vol. 111, Birkhäuser–Verlag, Basel 1993.
M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193 (2000), no. 2, 419–461.
C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292(1992), no. 4, 685–710.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0