[Paper Review] Functors and Computations in Floer homology with Applications Part II

Claude Viterbo|arXiv (Cornell University)|May 3, 2018

Geometric and Algebraic TopologyMathematics4 references169 citations

TL;DR

The paper proves Floer cohomology is isomorphic to GF-homology and that FH*(DT*N) is isomorphic to H*(ΛN), via generating functions and gradient-flow analysis.

ABSTRACT

The results in this paper concern computations of Floer cohomology using generating functions. The first part proves the isomorphism between Floer cohomology and Generating function cohomology introduced by Lisa Traynor. The second part proves that the Floer cohomology of the cotangent bundle (in the sense of Part I), is isomorphic to the cohomology of the loop space of the base. This has many consequences, some of which were given in Part I (GAFA, Geom. funct. anal. Vol. 9 (1999) 985-1033), others will be given in forthcoming papers. The results in this paper had been announced (with indications of proof) in a talk at the ICM 94 in Z{ü}rich. Up to typos, this is the revised version from 2003.

Motivation & Objective

• Motivate computations of Floer cohomology using generating functions.
• Establish an isomorphism between Floer cohomology and GF-homology for Lagrangian pairs.
• Show that the Floer cohomology of the cotangent bundle DT*N equals the loop space cohomology H*(ΛN).
• Extend Part I results and set groundwork for applications to loop space topology.

Proposed method

• Define and use a generating function S with quadratic at infinity for Lagrangian submanifolds.
• Construct A_H to interpolate between action A and generating function S, and define FH*(L0,L1; a,b) via critical points and Floer trajectories.
• Prove FH*(L0,L1; a,b) ≃ GF*(L0,L1; a,b) by comparing Floer complexes with GF cohomology.
• Relate DT*N Floer cohomology to H*(ΛN) through a sequence of isomorphisms involving diagonal submanifolds, graphs of symplectomorphisms, and Conley index arguments.
• Employ gradient-like flows and almost complex structures to link Floer trajectories with Morse theoretic trajectories of generating functions.
• Utilize Legendre duality and action functional analysis to connect Hamiltonian Floer theory with loop space functionals.

Experimental results

Research questions

• RQ1 Does Floer cohomology FH*(L0,L1; a,b) coincide with GF*(L0,L1; a,b) for Lagrangian pairs with generating functions quadratic at infinity?
• RQ2 Can FH*(DT*N) be identified with the cohomology H*(ΛN) of the base's loop space?
• RQ3 What are the intermediate isomorphisms and constructions (e.g., diagonal trick, graph押) that connect Floer theory to generating function and loop space cohomology?
• RQ4 How do gradient-flow perturbations and Conley index techniques illuminate the Floer–GF–loop space equivalence?

Key findings

• FH*(L0,L1; a,b) is isomorphic to GF*(L0,L1; a,b) for generating functions quadratic at infinity.
• For cotangent bundles, FH*(DT*N) is isomorphic to H*(ΛN), the cohomology of the loop space of N.
• The isomorphism framework extends to S^1-equivariant cohomology with rational coefficients.
• The paper provides a detailed construction showing that Floer trajectories correspond to gradient trajectories of the generating function S, via interpolation A_H and suitable almost complex structures.