241206 Floer homology and the Seiberg-Witten equations II (V) (WiSe 2023/2024)

This is the second part of 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 and how they naturally arise from the attempt to define Seiberg-Witten invariants for 4-manifolds with boundary.

Having discussed Morse-Floer homology and Conley index theory in finite dimensional situations in the first part of the course, we try to apply the ideas to the infinite dimensional situation that arises from studying the Seiberg-Witten equations on cylinders RxY where Y is a closed spin^c 3-manifold. We focus on the constructions of monopole Floer homology by Kronheimer and Mrowka and the Seiberg-Witten-Floer homotopy type by Manolescu.

We then shift our focus back to dimension 4. Suitable 4-dimensional cobordisms from one 3-manifold to another will induce maps between Floer homology groups and 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

This is the second part of a lecture series. All the material discussed in the first part will be taken for granted. This includes Morse-Floer homology, Conley index theory, spin geometry, and the construction of the Seiberg-Witten invariants for closed 4-manifolds.

Bibliography

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