241116 Floer homology and the Seiberg-Witten equations (V) (SoSe 2023)

This is an advanced course about smooth manifolds of dimensions 3 and 4. The goal is to explain the constructions of monopole Floer homology and Seiberg-Witten-Floer homotopy types for 3-manifolds in as much detail as time permits. There will be accompanying exercise classes with weekly homework assignments. The course is divided into three parts:

The first part begins with a review of Floer homology in the finite dimensional context, which computes the (ordinary) homology of a manifold from the gradient flow of a Morse function. We then extend and refine this to the Conley index, which associates to an isolated invariant set of a flow a homotopy type. These construction will serve as blueprints later.

In the second part, we introduce spin^c structures and the Seiberg-Witten equations on 3- and 4-manifolds. We show that the Seiberg-Witten equations on 3-manifolds appear formally as a gradient flow equation of a function on an infinite dimensional space. While the ideas of Floer homology and Conley index theory cannot be applied directly, some ideas can be carried over leading to powerful invariants of 3-manifolds. We focus on the constructions of monopole Floer homology by Kronheimer and Mrowka and the Seiberg-Witten-Floer homotopy type by Manolescu.

In the final part, we focus on the relation to 4-manifolds. Suitable 4-dimensional cobordisms from one 3-manifold to another will induce maps between Floer homology groups and SWF homotopy types which reflect certain topological properties of the cobordism. As an application, we will obtain restrictions on the possible intersection forms of 4-manifolds with prescribed boundary. A consequence of this is the existence of topological 4--manifolds which do not admit smooth structures.

Requirements for participation, required level

Necessary: basic knowledge of Morse functions and their gradient flows, homology and cohomology theory (e.g. singular, cellular, simplicial, etc), basic homotopy theory (homotopy sets, long exact sequences for fibrations and cofibrations),
Helpful: basics of spin geometry (Clifford algebras, spin groups, spin structures, Dirac operators), Sobolev spaces (of sections of vector bundles over a manifold)

Bibliography

Nicolaescu - An Invitation to Morse Theory (Springer, Universitext, 2011, 2nd ed.)
Salamon - Connected simple systems and the Conley index of isolated invariant sets (Trans. Amer. Math. Soc. 291 (1985), no. 1, 1-41)
Kronheimer, Mrowka - Monopoles and Three-Manifolds (Cambridge University Press, 2008)
Manolescu - Seiberg-Witten-Floer stable homotopy type of three-manifolds with b_1=0 (Geom. Topol. 7 (2003), 889-932)

External comments page

https://www.math.uni-bielefeld.de/~sbehrens/Floer23.html