On the Floer homology of cotangent bundles

By Alberto Abbondandolo, M. Schwarz. 2004

244 citations recorded by the source corpus.

This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle T* M of a compact orientable manifold M. The first result is a new L∞ estimate for the solutions of the Floer equation, which allows us to deal with a larger—and more natural—class of Hamiltonians. The second and main result is a new construction of the isomorphism between the Floer homology and the singular homology of the free loop space of M in the periodic case, or of the based loop space of M in the Lagrangian intersection problem. The idea for the construction of such an isomorphism is to consider a Hamiltonian that is the Legendre transform of a Lagrangian on T M and to construct an isomorphism between the Floer complex and the Morse complex of the classical Lagrangian action functional on the space of W1,2 free or based loops on M. © 2005 Wiley Periodicals, Inc.

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Citations in the corpus

The homology of path spaces and Floer homology with conormal boundary conditions (2008, unknown): “1 in [3] for more details on the arguments just sketched When i(γ) = μ(x), compactness and transversality imply that M (γ, x) is a finite set.”
Topological quantum field theory structure on symplectic cohomology (2010, background): “Abbondandolo-Schwarz [1] proved that the products on HW (T ∗ qN ⊂ T N) and SH(T N) agree with the Pontrjagin and Chas-Sullivan products on Hn−∗(ΩN) and Hn−∗(LN) via the respective isomorphisms.”
The Gysin exact sequence for $S^1$-equivariant symplectic homology (2009, methodology): “…diagram whose rows and columns are, respectively, the tautological and Gysin exact sequences ... ... ... ... . . . // SH+k+1 // Hk+n // SHk // SH+k // . . . . . . // SH+,S 1 k+1 // HS 1 k+n // SHS 1 k // SH+,S 1 k // . . . . . . // SH+,S 1 k−1 // HS 1 k+n−2 // SHS 1 k−2 // SH+,S 1 k−2 // .…”
Loop Products and Closed Geodesics (2007, methodology): “1.6. There is a well-known isomorphism between the Floer homology of the cotangent bundle of M and the homology of the free loop space of M, which transforms the pair-ofpants product into the Chas-Sullivan product on homology, see [ AS1 , AS2, SaW, Vi, CHV].”
An open string analogue of Viterbo functoriality (2007, background): “We emphasize again that, even though the cochain level construction itself is new, the underlying cohomology level theory is well-known to specialists, as one will see by looking at [1].”
An exact sequence for contact- and symplectic homology (2007, methodology): “The above isomorphism actually holds in an improved version involving, on the left hand side, the symplectic homology groups truncated by the values of the action functional and, on the right hand side, the relative homology groups of sublevel sets of the energy functional on the loop space [1, 25].”
Rabinowitz Floer homology and symplectic homology (2009, unknown): “Let w = (x, η) : R→ L×R be a solution of (26) whose asymptotic limits x± are contained in V ∪(M× [1, 2)).”
Contact geometry of the restricted three‐body problem (2010, result): “Combining this with the work of Viterbo [19], Salamon-Weber [18], Abbondandolo-Schwarz [1] SH ∗ ( T ∗≤ 1 S 2 ) ∼ = H ∗ (Λ S 2 ) where Λ S 2 denotes the free loop space of S 2 we see that in both cases, c H ( L 1 ) and c H ( L 1 )+ (cid:15) , Rabinowitz Floer homology of the energy hypersurface…”
On the wrapped Fukaya category and based loops (2009, methodology): “This result relies on the fact that a certain map we shall construct is an inverse to an isomorphism constructed by Abbondandolo and Schwarz in [1].”
Symplectic cohomology and Viterbo's theorem (2013, methodology): “More speculatively, one could attempt to implement the idea of [3] directly in the framework of linear Hamiltonians.”
Lagrangian cobordisms via generating families: Construction and geography (2014, unknown): “Lagrangian cobordisms between Legendrian submanifolds have recently enjoyed increasing interest, partly due to their centrality in symplectic field theory (see Eliashberg, Givental and Hofer [19], and see also Bourgeois, Ekholm and Eliashberg [7] and Ekholm [13; 14] for recent examples) and wrapped ”
Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors (2017, methodology): “hain complexes, and the A∞ operations are deﬁned by counting perturbed holomorphic polygones with Lagrangian boundary conditions. Wrapped Floer cohomology was deﬁned by A. Abbondandolo and M. Schwarz [1], at least for cotangent ﬁbres, but the general deﬁnition and the chain level construction needed”
Erratum to “Geometry of contact transformations and domains: orderability versus squeezing” (2005, background): “53D10, 53D40; 53D35, 53D50 Dedicated to Dusa McDuff on the occasion of her 60th birthday”