IAS/Park City Mathematics Series
Volume 7, 1999Lectures on Floer Homology
Dietmar Salamon
Introduction
The purpose of these lectures is to give an introduction to symplectic Floer
homology and the proof of the Arnold conjecture. This conjecture gives a lower
bound for the number of 1-periodic solutions of a 1-periodic Hamiltonian system in
terms of the sum of the Betti numbers.
The first three lectures are introductory, and deal with the basic ideas in Floer's
proof of the Arnold conjecture. Topics covered include the Morse-Smale-Witten
complex, some basic analysis and Fredholm theory, the spectral flow and the Maslov
index, the compactness problem, the construction of Floer homology, the proof that
Floer homology is an invariant, and the role of Novikov rings.
The last two lectures deal with more recent developments and lead up to a proof
of the Arnold conjecture for general symplectic manifolds and rational coefficients.
The fourth lecture gives an introduction to Gromov compactness, stable maps, and
the Deligne-Mumford compactification, while the last lecture discusses multi-valued
perturbations, branched manifolds, the construction of rational Gromov-Witten
invariants, and the proof of the Arnold conjecture for general symplectic manifolds.(^1) Math Institute, University of Warwick, Coventry CV4 7AL, United Kingdom.
E-mail address: das©m,aths. 11ar11ick. ac. uk.
Diagram courtesy of Joachim Web er , used with gratitude from the author.
@19 99 America n Mathematical Society
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