Abstract

Given a closed smooth manifold (and an appropriate Morse function and metric) you can define the Morse cochain complex, whose cohomology is isomorphic to that of the usual singular cochain complex. You can also define a product on the Morse complex, which induces the familiar cup product on cohomology, but in general it fails to be associative at chain level and does not encode all of the structure contained in the singular complex (e.g. Massey products). I will describe how an idea of Fukaya leads naturally to the notion of an A-infinity algebra, which is the correct weakening of the notion of associativity, and a way to build the structure of such an algebra on the Morse complex so that it captures (essentially) all of the information of the singular complex. If time permits I will also discuss how to quantise (i.e. deform) this algebra in certain ways.