A Symplectic Prolegomenon 417
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, Number 3, July 2015, Pages 415–464 S 0273-0979(2015)01477-1 Article electronically published on January 30, 2015 ASYMPLECTICPROLEGOMENON IVAN SMITH Abstract. A symplectic manifold gives rise to a triangulated A∞-category, the derived Fukaya category, which encodes information on Lagrangian sub- manifolds and dynamics as probed by Floer cohomology. This survey aims to give some insight into what the Fukaya category is, where it comes from, and what symplectic topologists want to do with it. ...everything you wanted to say required a context. If you gave the full context, people thought you a rambling old fool. If you didn’t give the context, people thought you a laconic old fool. Julian Barnes, Staring at the Sun 1. Introduction The origins of symplectic topology lie in classical Hamiltonian dynamics, but it also arises naturally in algebraic geometry, in representation theory, in low- dimensional topology, and in string theory. In all of these settings, arguably the most important technology deployed in symplectic topology is Floer cohomol- ogy, which associates to a pair of oriented n-dimensional Lagrangian submanifolds L, L of a 2n-dimensional symplectic manifold (X, ω)aZ2-graded vector space HF∗(L, L), which categorifies the classical homological intersection number of L and L in the sense that χ(HF∗(L, L)) = (−1)n(n+1)/2 [L] · [L]. In the early days of the subject, Floer cohomology played a role similar to that taken by singular homology groups in algebraic topology at the beginning of the last century: the ranks of the groups provided lower bounds for problems of geo- metric or dynamical origin (numbers of periodic points, for instance).
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