Morse theory for Lagrangian intersections
type: Item
instance of: scholarly article
title: Morse theory for Lagrangian intersections
zbMATH Open document ID: 0674.57027
published in: Journal of Differential Geometry
publication date: 1988-01-01T00:00:00Z
review text: This important paper develops ``Floer cohomology'' in a symplectic context, with the aim of proving the Arnold conjecture about the number of fixed points of an exact symplectic diffeomorphism. The author's abstract: ``Let P be a compact symplectic manifold and let \(L\subset P\) be a Lagrangian submanifold with \(\pi_ 2(P,L)=0\). For any exact diffeomorphism \(\phi\) of P with the property that \(\phi\) (L) intersects L transversally, we prove a Morse inequality relating the set \(\phi\) (L)\(\cap L\) to the cohomology of L. As a consequence, we prove a special case of the Arnold conjecture: If \(\pi_ 2(P)=0\) and \(\phi\) is an exact diffeomorphism all of whose fixed points are non-degenerate, then the number of fixed points is greater than or equal to the sum over the \({\mathbb{Z}}_ 2\)-Betti numbers of P.'' The Morse inequality comes from the ``Floer cohomology'' of a complex \(I^*(L,\phi,J)\) where \(I^ p(L,\phi,J)\) is the \({\mathbb{Z}}_ 2\)-module generated by the points in \(\phi\) (L)\(\cap L\) which have ``index p''. In order to define the boundary operator of \(I^*\) he observes that these intersection points are the critical points of a certain action functional \({\mathfrak a}\) on a space \(\Omega\) of paths z(t) in P which join a point z(0)\(\in L\) to a point z(1)\(\in \phi (L)\). Floer chooses a metric on \(\Omega\) using an auxiliary almost complex structure J on P. (If P were Kähler, this could just be the Kähler metric.) This has the property that the trajectories \(z_{\tau}\) of the gradient flow of \({\mathfrak a}\) on \(\Omega\) with respect to this metric fit together to form a J-holomorphic map u: (\(\tau\),t)\(\mapsto z_{\tau}(t)\) of the strip \({\mathbb{R}}\times [0,1]\) into P. Even though the gradient flow of \({\mathfrak a}\) is not globally defined on \(\Omega\), the trajectories which go from one critical point to another are well-behaved for generic J because they are solutions of an elliptic p.d.e.. Therefore, one can define the …
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 57R70
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 57R42
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 58J10
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 4104008
zbMATH Keywords: Lagrangian intersection
zbMATH Keywords: index of intersection points
zbMATH Keywords: elliptic operators
zbMATH Keywords: Floer cohomology
zbMATH Keywords: number of fixed points of an exact symplectic diffeomorphism
zbMATH Keywords: compact symplectic manifold
zbMATH Keywords: Lagrangian submanifold
zbMATH Keywords: Morse inequality
zbMATH Keywords: almost complex structure
zbMATH Keywords: gradient flow
author: Andreas Floer
reviewed by: Dusa McDuff
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214442477
OpenAlex ID: W1800967450
Wikidata QID: Q63285610
DOI: 10.4310/JDG/1214442477
description: scientific article; zbMATH DE number 4104008