arXiv subjects
Publications and source records attributed to Stefan Behrens.
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.
We show that a four-manifold admits a boundary Lefschetz fibration over the disc if and only if it is diffeomorphic to $S^1 \times S^3\# n \overline{\mathbb{C} P^2}$, $\# m\mathbb{C} P^2 \#n\overline{\mathbb{C} P^2}$ or $\# m (S^2 \times S^2)$. Given the relation between boundary Lefschetz fibrations and stable generalized complex structures, we conclude that the manifolds $S^1 \times S^3\# n \overline{\mathbb{C} P^2}$, $\#(2m+1)\mathbb{C} P^2 \#n\overline{\mathbb{C} P^2}$ and $\# (2m+1) S^2 \times S^2$ 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.
We use Heegaard Floer homology with twisted coefficients to define numerical invariants for arbitrary closed 3-manifolds equipped torsion spin$^c$ 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 ${\rm HF}^\infty$. 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.
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.
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.