Description

Algebraic topology provide certain abelian groups, the singular homology groups, which are associated to a manifold. These allow to say, for instance, that two given manifolds cannot be the same (up to a suitable notion of isomorphism). On a manifold, one can study functions and its critical points - points where their derivative vanishes. Interestingly, the Hessian of such a function at a critical point provides topological information about the manifold. Furthermore, the gradient vector field of the function has flow lines which relate critical points. By suitably counting these flow lines, some interesting abelian groups can be constructed, the Morse homology groups . For a compact manifold with boundary, this Morse homology contains the same information as the singular homology groups.

A suitable analogue on some infite dimensional manifolds has been introduced by Andreas Floer, leading to the concept of Floer homology . This theory, starting in the 1980s, has been developped in various flavours, and has revolutionised the topology of 3- and 4-dimensional manifolds, and symplectic topology. Today, it still is an active research area.

Prerequisites

Co-requisites

Resources

• Morse homology
• Floer homology

Relevant books/lecturee notes include the following:

• Morse Theeory and Floer homology, Michèle Audin, Mihai Damian, Springer Universitytext (this reference is most detailed and self-contained)
• The moduli space of gradient flow lines and Morse homology, Urs Frauenfelder, Robert Nicholls, available online here .

The first book is available online in the Library for instance here. (There is also an english translation of this book)