Generalizations of planar contact manifolds to higher dimensions

Abstract

Iterated planar contact manifolds are a generalization of three dimensional planar contact manifolds to higher dimensions. We study some basic topological properties of iterated planar contact manifolds and discuss several examples and constructions showing that many contact manifolds are iterated planar. We also observe that for any odd integer m > 3, any finitely presented group can be realized as the fundamental group of some iterated planar contact manifold of dimension m. Moreover, we introduce another generalization of three dimensional planar contact manifolds that we call projective. Finally, building symplectic cobordisms via open books, we show that some projective contact manifolds admit explicit symplectic caps.

Figures (4)

• Figure 1: The $4$--manifold $W_k$ shown on the left. The $2$--handle runs $n$ times over the $1$--handle. On the right, is the $3$--manifold boundary of $W_k$. The thin red curve is a regular fiber in the fibration and has contact framing $0$ with respect to the fibration framing. All surgery coefficients are with respect to the topological framings, not contact framings.
• Figure 2: A surgery diagram for $\partial W_{k}$. All surgery coefficients are with respect to the contact framing.
• Figure 3: The Stein domain $X_k$.
• Figure 4: The "handle" attachment. The green on the left is $X \times D^2$. The red lines represent the flow lines of the Liouville field $Z$. The blue is the smoothed boundary of the handle we want to attach. The actual handle is depicted on the right.

Theorems & Definitions (58)

• Theorem 1.1: Acu 2017, Acu17preprint
• Theorem 1.2: Acu-Moreno 2018, AcuMoreno18preprint
• Theorem 1.3: Acu-Moreno 2018, AcuMoreno18preprint
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.10
• Proposition 1.11