Floer homology of cotangent bundles and the loop product

By Alberto Abbondandolo, M. Schwarz. 2008

135 citations recorded by the source corpus.

We prove that the pair-of-pants product on the Floer homology of the cotangent bundle of a compact manifold M corresponds to the Chas–Sullivan loop product on the singular homology of the loop space of M . We also prove related results concerning the Floer homological interpretation of the Pontrjagin product and of the Serre fibration. The techniques include a Fredholm theory for Cauchy–Riemann operators with jumping Lagrangian boundary conditions of conormal type, and a new cobordism argument replacing the standard gluing technique.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1734e373b9dc7dbb8be6892fda434f468fbd2f0a

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In Mathematical Physics

Citations in the corpus

Lusternik–Schnirelmann theory and closed Reeb orbits (2016, methodology): “th ﬁnite stabilizer. A Floer theoretic analog of Lemma 2.3 for closed symplectic manifolds has been proved in [How12, How13] by using the pair-of-pants product. With the deﬁnition of the product from [AS], the argument readily extends to admissible Hamiltonians on the symplectic completion of a Liou”
Topological quantum field theory structure on symplectic cohomology (2010, background): “For a disc cotangent bundle of a closed manifold, SH∗(DT ∗N) is the homology of the free loop space LN=C∞(S1, N), and Abbondandolo-Schwarz [2] proved that SH∗(DT ∗N) has a “pair-of-pants product” which turns into the Chas-Sullivan loop product of string topology [9] on H∗(LN).”
A Beginner’s Introduction to Fukaya Categories (2013, background): “This and other related results can be viewed as the culmination of over a decade of investigations of the deep connections between the symplectic topology of T ∗N and the algebraic topology of the loop space of N , as previously studied by Viterbo [48], Salamon-Weber [39], Abbondandolo-Schwarz [1, 2”
The monotone wrapped Fukaya category and the open-closed string map (2012, background): “M = A[1] has composition maps μ r|s A[1] = μ r+1+s A .”
Sectorial descent for wrapped Fukaya categories (2018, background): “Given two Lagrangians L,K ⊆ X together with a short Reeb chord γ from ∂∞L to ∂∞K, we denote by L#γK the result of attaching this cobordism at infinity (more precisely, to define this surgery requires a certain Darboux chart, which contains an obvious Reeb chord γ).”
A Smooth Pseudo-Gradient for the Lagrangian Action Functional (2008, background): “However, in other cases it is useful to consider more general Lagrangian functions, for instance because this allows to achieve the transversality needed in Floer theory without perturbing the Levi-Civita almost complex structure on T M , as in [6].”
On the wrapped Fukaya category and based loops (2009, methodology): “8 of [2] implies that for a generic choice of almost complex structures the evaluation map H(x)→ Ω(1,2)(q, q) is transverse to ascending manifolds of S of codimension deg(x) or higher for every X-chord x.”
Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization (2011, unknown): “onventions amounts to applying the Poincar´e duality in each variable, which transforms the cup product on cohomology into the intersection product on homology. Now, a computation shows (compare with [AS]) that if u∈M(γ,γ′;γ′′), then AH(γ) +AH′ (γ′)−AH′′ (γ′′) ≥E(u)−ε, where E(u) ≥0 is the energy of”
Spectral invariants for monotone Lagrangians (2015, methodology): “ERCQ, FROL ZAPOLSKY sharp inequality was proved by Schwarz [Sch00]. A sharp triangle inequality for Lagrangian spectral invariants was proved in [MVZ12] based on the trick due to Abbondandolo–Schwarz [AS10]. Theorem 5 was proved by Leclercq [Lec08]for weakly exact M. The canonical isomorphism QH∗(∆)”
Nearby Lagrangians with vanishing Maslov class are homotopy equivalent (2010, methodology): “4) V0 ⊗ S → V1[1]⊗ S of degree 1, which is determined by a map (A.”
The homology of path spaces and Floer homology with conormal boundary conditions (2008, methodology): “Other choices are also interesting: for instance in [4] it is shown that the pair-of-pants product on Floer homology for periodic orbits on the cotangent bundle of M factors through a Floer homology for Hamiltonian orbits on T (M ×M) with nonlocal boundary condition (27) given by the submanifold Q o”