Stefan Behrens
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About me
I was an assistant professor (wissenschaftlicher Assistent) in the Geometry and Topology group led by Prof. Dr. Stefan Bauer from 2017 to 2024. This homepage will eventually find a new home. My field of research is low dimensional topology with a focus on the topology of smooth 4-manifolds. Before coming to Bielefeld, I had stops in Bonn, Budapest, and Utrecht.
Research Interests
I am interested Seiberg-Witten and Heegaard-Floer theory and their application to 4-manifold theory. Recently, I have been thinking about various incarnations of so called correction terms (or d-invariants or Froyshov invariants) and what they have to say about intersection forms of 4-manifolds with non-empty boundary. I am currently thinking about various Froyshov-type invariants that arise in the context of Manolescu's Seiber-Witten-Floer homotopy types.
Here's a list of my mathematical interests:
- The topology of 4-Manifolds (mainly smooth but also topological)
- Heegaard-Floer and Seiberg-Witten theory (both theory and applications)
- Stable homotopy theory (with a view toward applications to Seiberg-Witten theory)
- (Broken) Lefschetz fibrations and related creatures (wrinkled fibrations, Morse 2-functions, ...)
Published Articles & Preprints
The monopole h-invariants from a topological perspective
preprint, submitted
preprint, submitted
We study the monopole h-invariants of 3-manifolds from a topological perspective based on Lidman and Manolescu's description of monopole Floer homology in terms of Seiberg-Witten-Floer homotopy types. We investigate the possible dependence on the choice of coefficients and give proofs of several properties of the h-invariants which are well known to experts, but hard to track down in the literature.
Classification of boundary Lefschetz fibrations over the disc
(with G. R. Cavalcanti and R. L. Klaasse)
"Geometry and Physics: A Festschrift in Honour of Nigel Hitchin, Volume II", Oxford University Press 2018
"Geometry and Physics: A Festschrift in Honour of Nigel Hitchin, Volume II", Oxford University Press 2018
We show that a four-manifold admits a boundary Lefschetz fibration over the disc if and only if it is diffeomorphic to S1×S3#n-CP2, #mCP2#n-CP2 or #m(S2×S2). Given the relation between boundary Lefschetz fibrations and stable generalized complex structures, we conclude that the manifolds S1×S3#n-CP2, #(2m+1)CP2#nCP2 and #(2m+1)S2×S2 admit stable structures whose type change locus has a single component and are the only four-manifolds whose stable structure arise from boundary Lefschetz fibrations over the disc.
Heegaard Floer correction terms, with a twist
(with M. Golla)
Quantum Topol., Vol. 9 (2018), No. 1, 1-37
Quantum Topol., Vol. 9 (2018), No. 1, 1-37
We use Heegaard Floer homology with twisted coefficients to define numerical invariants for arbitrary closed 3-manifolds equipped torsion spinc structures, generalising the correction terms (or d-invariants) defined by Ozsváth and Szabó for integer homology 3-spheres and, more generally, for 3-manifolds with standard HF-infinity. Our twisted correction terms share many properties with their untwisted analogues. In particular, they provide restrictions on the topology of 4-manifolds bounding a given 3-manifold.
Elimination of cusps in dimension 4 and its applications
(with K. Hayano)
Proc. Lond. Math. Soc., Vol. 113 (2016), No. 5, 674-724
Proc. Lond. Math. Soc., Vol. 113 (2016), No. 5, 674-724
Several new combinatorial descriptions of closed 4-manifolds have recently been introduced in the study of smooth maps from 4-manifolds to surfaces. These descriptions consist of simple closed curves in a closed, orientable surface and these curves appear as so called vanishing sets of corresponding maps. In the present paper we focus on homotopies canceling pairs of cusps so called cusp merges. We first discuss the classification problem of such homotopies, showing that there is a one-to-one correspondence between the set of homotopy classes of cusp merges canceling a given pair of cusps and the set of homotopy classes of suitably decorated curves between the cusps. Using our classification, we further give a complete description of the behavior of vanishing sets under cusp merges in terms of mapping class groups of surfaces. As an application, we discuss the uniqueness of surface diagrams, which are combinatorial descriptions of 4-manifolds due to Williams, and give new examples of surface diagrams related with Lefschetz fibrations and Heegaard diagrams.
On 4-Manifolds, Folds and Cusps
Pacific J. Math., Vol. 264 (2013), No. 2, 257-306
Pacific J. Math., Vol. 264 (2013), No. 2, 257-306
We study simple wrinkled fibrations, a variation of the simplified purely wrinkled fibrations introduced by Williams, and their combinatorial description in terms of surface diagrams. We show that simple wrinkled fibrations induce handle decompositions on their total spaces which are very similar to those obtained from Lefschetz fibrations. The handle decompositions turn out to be closely related to surface diagrams and we use this relationship to interpret some cut-and-paste operations on 4-manifolds in terms of surface diagrams. This, in turn, allows us classify all closed 4-manifolds which admit simple wrinkled fibrations of genus one, the lowest possible fiber genus.
Books
- The Disk Embedding Theorem (joint with Boldizsar Kalmar, Min Hoon Kim, Mark Powell, and Arunima Ray) (Oxford University Press, 2021)
Theses
- Smooth 4-Manifolds and Surface Diagrams (Doctoral dissertation, University of Bonn, 2014)
- The L²-Stokes Theorem on certain incomplete manifolds (Diplom thesis, University of Bonn, 2009)
Teaching / Lehre
- Geometrie und Topologie II (WiSe 2020/2021)
- Geometrie und Topologie (SoSe 2020)
- Differentialtopologie (WiSe 2019/2020)
- Seminar zur Morse Theorie (WiSe 2019/2020)
- Proseminar über Knotentheorie (SoSe 2019)
- Proseminar über gewöhnliche Differentialgleichungen (WiSe 2018/2019)
Last update: December 8, 2024