Abstract
Given the cotangent bundle T*Q of a smooth manifold with its canonical symplectic structure, and a Hamiltonian function on T*Q which is fiberwise asymptotically quadratic, its well-defined Floer homology with the pair-of-pants ring structure is ring-isomorphic to the singular homology of the free loop space of Q endowed with its loop product. The analogous statement is true for the based loop space versions and the Pontrjagin product. This article gives an overview of the construction of this ring isomorphism which is based on Legendre duality and moduli spaces of flow trajectories of hybrid type, which are half Floer trajectories for the Hamiltonian problem and half Morse trajectories for the Lagrangian problem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abbondandolo, A. and Majer, P. (2005) Lectures on the Morse complex for infinite-dimensional manifolds, in this volume.
Abbondandolo, A. and Schwarz, M. (2004) On the Floer homology of cotangent bundles, Comm. Pure Appl. Math., to appear arXiv:math. SG/0408280.
Abbondandolo, A. and Schwarz, M. (2005) Floer homology of cotangent bundles and the loop product, in preparation.
Benci, V. (1986) Periodic solutions of Lagrangian systems on a compact manifold, J. Differential Equations 63, 135–161.
Chas, M. and Sullivan, D. (1999) String topology, arXiv:math. GT/9911159.
Cohen, R. (2005) Morse theory, graphs, and string topology, in this volume.
Cohen, R. and Jones, J. D. S. (2002) A homotopy theoretic realization of string topology, Math. Ann. 324, 773–798.
Cohen, R. L. and Voronov, A. A. (2005) Notes on string topology, arXiv:math. GT/0503625.
Duistermaat, J. J. (1976) On the Morse index in variational calculus, Adv. in Math. 21, 173–195.
Floer, A. (1988)a A relative Morse index for the symplectic action, Comm. Pure Appl. Math. 41, 393–407.
Floer, A. (1988)b The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41, 775–813.
Floer, A. (1989) Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120, 575 –611.
Floer, A. and Hofer, H. (1993) Coherent orientations for periodic orbit problems in symplectic geometry, Math. Z. 212, 13–38.
Félix, Y., Halperin, S., and Thomas, J.-C. (2001) Rational Homotopy Theory, Vol. 205 of Grad. Texts in Math., New York, Springer.
Gromov, M. (1985) Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82, 307 –347.
Hofer, H. and Salamon, D. A. (1995) Floer homology and Novikov rings, In H. Hofer, C. H. Taubes, A. Weinstein, and E. Zehnder (eds.), The Floer Memorial Volume, Vol. 133 of Progr. Math, pp. 483–524, Basel, Birkhäuser.
Piunikhin, S., Salamon, D., and Schwarz, M. (1996) Symplectic Floer–Donaldson theory and quantum cohomology, In C. B. Thomas (ed.), Contact and Symplectic Geometry, Vol. 8 of Publ. Newton Inst., Cambridge 1994, pp. 171–200, Cambridge, Cambridge Univ. Press.
Ramirez, A. (2005), Ph. D. thesis, Stanford University, in preparation.
Robbin, J. and Salamon, D. (1993) The Maslov index for paths, Topology 32, 827–844.
Salamon, D. and Weber, J. (2003) Floer homology and the heat flow, arXiv:math. SG 0304383.
Schwarz, M. (1993) Morse Homology, Vol. 111 of Progr. Math, Basel, Birkhäuser.
Schwarz, M. (1995) Cohomology operations from S1 -cobordisms in Floer homology, Ph. D. thesis, ETH Zurich.
Viterbo, C. (1996) Functors and computations in Floer homology with applications. II, preprint.
Weber, J. (2002) Perturbed closed geodesics are periodic orbits : index and transversality, Math. Z. 241, 45–82.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer
About this paper
Cite this paper
ABBONDANDOLO, A., SCHWARZ, M. (2006). NOTES ON FLOER HOMOLOGY AND LOOP SPACE HOMOLOGY. In: Biran, P., Cornea, O., Lalonde, F. (eds) Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology. NATO Science Series II: Mathematics, Physics and Chemistry, vol 217. Springer, Dordrecht. https://doi.org/10.1007/1-4020-4266-3_02
Download citation
DOI: https://doi.org/10.1007/1-4020-4266-3_02
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-4272-0
Online ISBN: 978-1-4020-4266-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)