[Paper Review] On the Floer homology of cotangent bundles

Alberto Abbondandolo, Matthias Schwarz|ArXiv.org|Aug 20, 2004

Geometric and Algebraic Topology24 references4 citations

TL;DR

This paper establishes a new isomorphism between the Floer homology of the cotangent bundle $T^*M$ and the singular homology of the free loop space $Ω(M)$, using a Legendre-transformed Hamiltonian and a direct comparison with the Morse complex of the classical Lagrangian action functional. It introduces improved $L^\infty$ estimates for Floer trajectories, enabling a broader class of Hamiltonians and ensuring transversality and compactness via a novel geometric approach grounded in the cotangent bundle's fibration structure.

ABSTRACT

This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle of a compact orientable manifold M. The first result is a new uniform estimate for the solutions of the Floer equation, which allows 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 which is the Legendre transform of a Lagrangian on TM, and to construct an isomorphism between the Floer complex and the Morse complex of the classical Lagrangian action functional on the space of free or based loops on M of Sobolev class W(1,2).

Motivation & Objective

• To establish a canonical isomorphism between the Floer homology of $T^*M$ and the singular homology of the free loop space $\Omega(M)$, extending Viterbo's result with a direct geometric construction.
• To develop sharper $L^\infty$ estimates for solutions of the Floer equation on $T^*M$, valid for a wider class of Hamiltonians satisfying (H1) and (H2), independent of metric choices.
• To construct a chain-level isomorphism between the Floer complex of a Legendre-transformed Hamiltonian and the Morse complex of the classical action functional on $W^{1,2}$ loops.
• To prove transversality and compactness of moduli spaces of Floer trajectories using a novel gluing and perturbation scheme compatible with the Lagrangian foliation of $T^*M$.
• To ensure the isomorphism respects the decomposition of homology by conjugacy classes of $\pi_1(M)$, preserving the topological structure of loop spaces.

Proposed method

• Use of the Legendre transform to relate a Hamiltonian $H$ on $T^*M$ to a Lagrangian $L$ on $TM$, linking the Floer complex to the Morse complex of the action functional $\mathcal{E}$.
• Introduction of $L^\infty$ estimates for Floer trajectories via direct analysis of the Cauchy-Riemann operator, avoiding differentiation and the maximum principle.
• Construction of a chain map $\Theta_k: CM_k(\mathcal{E}) \to CF_k(H)$ defined by counting $J$-holomorphic curves $u \in \mathcal{M}^+(q,x)$ with signs $\epsilon(u)$.
• Application of the Sard-Smale theorem and Carleman similarity principle to prove residual regularity of the almost complex structure $J$ in $\mathcal{J}_{\mathrm{reg}}(H,g)$.
• Use of gluing theorems for broken trajectories to establish chain map properties, ensuring $\partial_{k-1}\Theta_k = \Theta_{k-1}\partial_k$.
• Proof that the matrix of $\Theta_k$ is lower triangular with $\pm 1$ on the diagonal under action-ordered bases, implying invertibility and thus a chain isomorphism.

Experimental results

Research questions

• RQ1 Can a direct isomorphism be constructed between the Floer homology of $T^*M$ and the singular homology of the free loop space $\Omega(M)$, without relying on generating functions or cohomological duality?
• RQ2 What $L^\infty$ bounds can be established for Floer trajectories on $T^*M$ under minimal assumptions on the Hamiltonian, independent of the Riemannian metric?
• RQ3 How can the Morse complex of the classical action functional on $W^{1,2}$ loops be canonically identified with the Floer complex of a Legendre-transformed Hamiltonian?
• RQ4 Under what conditions is the moduli space of Floer trajectories transverse and compact, ensuring well-defined Floer homology?
• RQ5 Does the isomorphism respect the decomposition of homology by conjugacy classes of $\pi_1(M)$, reflecting the topology of loop spaces?

Key findings

• The Floer homology $HF_*(T^*M)$ is canonically isomorphic to the singular homology of the free loop space $\Omega(M)$, with the isomorphism constructed via a chain-level correspondence between Morse and Floer complexes.
• The $L^\infty$ estimate for Floer trajectories is established under the natural conditions (H1) and (H2), which are metric-independent and allow for a broader class of Hamiltonians than previous results.
• The chain map $\Theta_k$ is shown to be invertible because its matrix representation is lower triangular with $\pm 1$ entries on the diagonal when generators are ordered by action.
• Transversality of the moduli spaces $\mathcal{M}^+(q,x)$ is achieved for residual $J$ in $\mathcal{J}_{\mathrm{reg}}(H,g)$, ensuring well-defined counts in the Floer differential.
• The isomorphism is compatible with the decomposition of homology by conjugacy classes of $\pi_1(M)$, reflecting the structure of the loop space fibration.
• The construction avoids the use of generating functions and cohomological methods, providing a direct geometric proof of the isomorphism, in contrast to Viterbo’s original approach.