[論文レビュー] Functors and Computations in Floer homology with Applications Part II

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

Geometric and Algebraic Topology参考文献 53被引用数 144

ひとこと要約

本論文は Floer cohomology が isomorphic to Generating Function cohomology であること、および FH*(DT*T N) が H*(ΛN) と同型であることを証明し、ループ空間と generating function 手法への含意を示す。

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{\\"u}rich. Up to typos, this is the revised version from 2003.

研究の動機と目的

• Motivate computing Floer cohomology via generating functions and establish isomorphism with GF-cohomology (as in Part I).
• Prove the Floer cohomology of cotangent bundles is isomorphic to the loop space cohomology of the base N.
• Explore consequences and potential alternative approaches to these isomorphisms.
• Extend the framework to equivariant settings with rational coefficients.
• Discuss foundational lemmas and constructions enabling these computations.

提案手法

• Define an interpolating functional A_H between the action A and a generating function S.
• Construct Floer cohomology FH^*(L0,L1; a,b) via counting J-holomorphic strips and relate it to GF^*(L0,L1; a,b).
• Prove FH^*(L0,L1; a,b) ≅ GF^*(L0,L1; a,b) using a family H and invariance arguments (Lemma 2.2).
• Show FH^*(O_N,L,0; a,b) ≅ H^*(S^b,S^a) when L is generated by a generating function S (Lemma 2.3).
• Relate FH^*(DT^*N) to H^*(ΛN) by diagonal arguments and Morse-type/Conley index techniques (Theorem 3.1).
• Employ Legendre duals and pseudogradient flows to connect Floer trajectories with gradient trajectories of the generated action S_Φ (Section 3).

実験結果

リサーチクエスチョン

• RQ1 Does FH^*(L0,L1; a,b) compute via GF cohomology for Lagrangians admitting a generating function?
• RQ2 Is FH^*(DT^*N) naturally isomorphic to H^*(ΛN), i.e., does Floer cohomology of the cotangent bundle recover loop space cohomology?
• RQ3 Can generating function cohomology provide alternative proofs or approaches to Floer cohomology computations?
• RQ4 What are the equivariant or rational-coefficient extensions of these isomorphisms?
• RQ5 How do the constructions relate to other approaches (e.g., gradient flow of geodesic energy) for proving the main theorems?

主な発見

• FH^*(L0,L1; a,b) is isomorphic to GF^*(L0,L1; a,b).
• FH^*(DT^*N) is isomorphic to H^*(ΛN).
• The isomorphisms extend to S^1-equivariant cohomology with rational coefficients.
• Generating functions provide a practical framework for computing Floer cohomology via Morse-theoretic data.
• The results unify Floer theory on cotangent bundles with loop space topology via a generating function formalism.