An Introduction to Floer Homology

Floer homology is a beautiful theory introduced in 1985 by Andreas Floer [8]. It combined new ideas about Morse theory, gauge theory, and Casson’s approach [1, 14] to homology 3-spheres and the representations of their fundamental groups into Lie groups such as SU(2) and SO(3). From its inception, it was related to the study of the anti-self-dual Yang-Mills equations on 4-manifolds, and is the receptacle for the relative Donaldson invariants of 4-manifolds with boundary [5]. Floer introduced two versions, one for Lagrangian submanifolds of a symplectic manifold, and another (Instanton Homology) for homology spheres. These threads were reunited with the introduction of Heegaard Floer homology some years later by Ozsváth and Szabó, and monopole homology by Kronheimer and Mrowka [10]. Even with these great advances, the instanton theory retains great interest due to its close connection with the fundamental group–the most basic invariant of a 3-manifold. Remarkable results in knot theory were proved by Kronheimer and Mrowka by developing versions of instanton homology for knots and links in a 3-manifold. There is still much to be learned from Floer’s original ideas! The plan for these two lectures is to briefly review the (ordinary) Morse background and to introduce basic notions about connections. Then we will see the definition and basic properties of the Chern-Simons invariant, and a sketch of the construction of the instanton homology of a homology 3-sphere. A second pair of lectures by Nikolai Saveliev will develop the basics of the instanton knot homology. The prerequisites are a general understanding of