9781402042720

Preface

Contributors

xiii

Lectures on the Morse Complex for Infinite-Dimensional Manifolds

(74)

A. Abbondandolo

P. Majer

A few facts from hyperbolic dynamics

(19)

Adapted norms

(2)

Linear stable and unstable spaces of an asymptotically hyperbolic path

(4)

Morse vector fields

(1)

Local dynamics near a hyperbolic rest point

(2)

Local stable and unstable manifolds

(1)

The Grobman - Hartman linearization theorem

(7)

Global stable and unstable manifolds

(2)

The Morse complex in the case of finite Morse indices

(31)

The Palais - Smale condition

(1)

The Morse - Smale condition

(1)

The assumptions

(1)

Forward compactness

(2)

Consequences of compactness and transversality

(2)

Cellular filtrations

(1)

The Morse complex

(5)

Representation of ∂ in terms of intersection numbers

(4)

How to remove the assumption (A8)

(1)

Morse functions on Hilbert manifolds

(2)

Basic results in transversality theory

(2)

Genericity of the Morse - Smale condition

(5)

Invariance of the Morse complex

(5)

The Morse complex in the case of infinite Morse indices

(23)

The program

(2)

Fredholm pairs and compact perturbations of linear subspaces

(1)

Finite-dimensional intersections

(3)

Essential subbundles

(2)

Orientations

(2)

Compactness

(4)

Two-dimensional intersections

(2)

The Morse complex

(2)

Bibliographical note

(6)

Notes on Floer Homology and Loop Space Homology

(34)

A. Abbondandolo

M. Schwarz

Introduction

(2)

Main result

(8)

Loop space homology

(3)

Floer homology for the cotangent bundle

(5)

Ring structures and ring-homomorphisms

(10)

The pair-of-pants product

(4)

The ring homomorphisms between free loop space Floer homology and based loop space Floer homology and classical homology

(6)

Morse-homology on the loop spaces ΛQ and ΩQ, and the isomorphism

(7)

Products in Morse-homology

102

(7)

Ring isomorphism between Morse homology and Floer homology

105

(4)

Homotopical Dynamics in Symplectic Topology

109

(40)

J.-F. Barraud

O. Cornea

Introduction

109

(1)

Elements of Morse theory

110

(17)

Connecting manifolds

113

(12)

Operations

125

(2)

Applications to symplectic topology

127

(22)

Bounded orbits

127

(3)

Detection of pseudoholomorphic strips and Hofer's norm

130

(19)

Morse Theory, Graphs, and String Topology

149

(36)

R. L. Cohen

Graphs, Morse theory, and cohomology operations

152

(13)

String topology

165

(8)

A Morse theoretic view of string topology

173

(5)

Cylindrical holomorphic curves in the cotangent bundle

178

(7)

Topology of Robot Motion Planning

185

(46)

M. Farber

Introduction

185

(1)

First examples of configuration spaces

186

(3)

Varieties of polygonal linkages

189

(1)

Short and long subsets

190

(1)

Poincare polynomial of M(a)

190

(4)

Universality theorems for configuration spaces

194

(1)

A remark about configuration spaces in robotics

195

(1)

The motion planning problem

195

(3)

Tame motion planning algorithms

198

(1)

The Schwarz genus

199

(1)

The second notion of topological complexity

200

(1)

Homotopy invariance

201

(1)

Order of instability of a motion planning algorithm

201

(1)

Random motion planning algorithms

202

(1)

Equality theorem

203

(4)

An upper bound for TC(X)

207

(1)

A cohomological lower bound for TC(X)

207

(1)

Examples

208

(1)

Simultaneous control of many systems

209

(1)

Another inequality relating TC(X) to the usual category

210

(1)

Topological complexity of bouquets

211

(1)

A general recipe to construct a motion planning algorithm

212

(1)

How difficult is to avoid collisions in Rm?

213

(1)

The case m = 2

214

(2)

TC(F(Rm, n)) in the case m ≥ 3 odd

216

(1)

Shade

217

(2)

Illuminating the complement of the braid arrangement

219

(1)

A quadratic motion planning algorithm in F(Rm, n)

220

(1)

Configuration spaces of graphs

221

(1)

Motion planning in projective spaces

222

(2)

Nonsingular maps

224

(2)

TC(RPn) and the immersion problem

226

(2)

Some open problems

228

(3)

Application of Floer Homology of Langrangian Submanifolds to Symplectic Topology

231

(46)

K. Fukaya

Introduction

231

(1)

Lagrangian submanifold of Cn

232

(3)

Perturbing Cauchy - Riemann equation

235

(5)

Maslov index of Lagrangian submanifold with vanishing second Betti number

240

(3)

Floer homology and a spectral sequence

243

(3)

Homology of loop space and Chas - Sullivan bracket

246

(5)

Iterated integral and Gerstenhaber bracket

251

(3)

A∞ deformation of de Rham complex

254

(5)

S1 equivariant homology of loop space and cyclic A∞ algebra

259

(1)

L∞ structure on H(S1 x Sn; Q)

260

(5)

Lagrangian submanifolds of C3

265

(1)

Aspherical Lagrangian submanifolds

266

(5)

Lagrangian submanifolds homotopy equivalent to S1 x S2m

271

(1)

Lagrangian submanifolds of CPn

272

(5)

The LS-index A Survey

277

(44)

M. Izydorek

Introduction

277

(5)

The LS-index

282

(8)

Basic definitions and facts

282

(3)

Spectra

285

(3)

The LS-index

288

(2)

Cohomology of spectra

290

(3)

Attractors, repellers and Morse decompositions

293

(3)

Equivariant LS-flows and the G-LS-index

296

(7)

Symmetries

296

(1)

Isolating neighbourhoods and the equivariant LS-index

296

(7)

Applications

303

(18)

A general setting

303

(2)

Applications of the LS-index

305

(4)

Applications of the cohomological LS-index

309

(3)

Applications of the equivariant LS-index

312

(9)

Lectures on Floer Theory and Spectral Invariants of Hamiltonian Flows

321

(96)

Y.-G. Oh

Introduction

321

(5)

The free loop space and the action functional

326

(12)

The free loop space and the S1-action in general

326

(1)

The free loop space of symplectic manifolds

327

(1)

The Novikov covering

328

(2)

Perturbed action functionals and their action spectra

330

(2)

The L2-gradient flow and perturbed Cauchy - Riemann equations

332

(4)

Comparison of two Cauchy - Riemann equations

336

(2)

Floer complex and the Novikov ring

338

(14)

Novikov - Floer chains and the Novikov ring

338

(3)

Definition of the Floer boundary map

341

(5)

Definition of the Floer chain map

346

(1)

Semi-positivity and transversality

347

(1)

Composition law of Floer's chain maps

348

(4)

Energy estimates and Hofer's geometry

352

(9)

Energy estimates and the action level changes

352

(3)

Energy estimates and Hofer's norm

355

(3)

Level changes of Floer chains under the homotopy

358

(1)

The ε-regularity type invariants

359

(2)

Definition of spectral invariants and their axioms

361

(6)

Floer complex of a small Morse function

361

(1)

Definition of spectral invariants

362

(3)

Axioms of spectral invariants

365

(2)

The spectrality axiom

367

(10)

A consequence of the nondegenerate spectrality axiom

368

(2)

Spectrality axiom for the rational case

370

(4)

Spectrality for the irrational case

374

(3)

Pants product and the triangle inequality

377

(9)

Quantum cohomology in the chain level

377

(3)

Grading convention

380

(2)

Hamiltonian fibrations and the pants product

382

(3)

Proof of the triangle inequality

385

(1)

Spectral norm of Hamiltonian diffeomorphisms

386

(13)

Construction of the spectral norm

386

(3)

The ε-regularity theorem and its consequences

389

(5)

Proof of nondegeneracy

394

(5)

Applications to Hofer geometry of Ham(M, ω)

399

(7)

Quasi-autonomous Hamiltonians and the minimality conjecture

399

(2)

Length minimizing criterion via ρ(H; 1)

401

(2)

Canonical fundamental Floer cycles

403

(1)

The case of autonomous Hamiltonians

404

(2)

Remarks on the transversality for general (M, ω)

406

(11)

Proof of the index formula

408

(9)

Floer Homology, Dynamics and Groups

417

(22)

L. Polterovich

Hamiltonian actions of finitely generated groups

417

(6)

The group of Hamiltonian diffeomorphisms

417

(1)

The no-torsion theorem

418

(2)

Distortion in normed groups

420

(1)

The No-Distortion Theorem

421

(1)

The Zimmer program

422

(1)

Floer theory in action

423

(5)

A brief sketch of Floer theory

423

(2)

Width and torsion

425

(1)

A geometry on Ham(M, ω)

425

(1)

Width and distortion

426

(1)

More remarks on the Zimmer program

426

(2)

The Calabi quasi-morphism and related topics

428

(11)

Extending the Calabi homomorphism

428

(1)

Introducing quasi-morphisms

429

(1)

Quasi-morphisms on Ham(M, ω)

430

(1)

Distortion in Hofer's norm on Ham(M, ω)

431

(2)

Existence and uniqueness of Calabi quasi-morphisms

433

(1)

``Hyperbolic'' features of Ham(M, ω)?

434

(1)

From π1 (M) to Diffo(M, Ω)

435

(4)

Symplectic topology and Hamilton - Jacobi equations

439

(22)

C. Viterbo

Introduction to symplectic geometry and generating functions

439

(8)

Uniqueness and first symplectic invariants

446

(1)

The calculus of critical level sets

447

(7)

The case of GFQI

450

(3)

Applications

453

(1)

Hamilton - Jacobi equations and generating functions

454

(2)

Coupled Hamilton - Jacobi equations

456

(5)

Index

461

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Morse Theoretic Methods in Nonlinear Analysis And in Symplectic Topology

1402042728

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