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Tight Contact Structures on Contact Mapping Tori and their Folded Sums

M. Firat Arikan

TL;DR

The paper develops a bundle-theoretic framework for folded sums of contact mapping tori and proves the existence of a co-oriented folded-contact structure η compatible with the natural circle fibration, with $dσ$ folded-symplectic on each fiber. In odd dimensions $2n+1≥7$, η is tight provided the induced boundary Reeb dynamics has no contractible periodic orbit, strengthening prior notions of tightness for folded constructions. The authors provide explicit bundle-based constructions via $σ = λ_β + K\,π^{*}(dθ)$ and a gluing-based folded sum approach, illustrating with the folded sum of standard bundles and other Weinstein/Liouville settings. These results broaden the toolkit for constructing tight contact structures on complex fibrations and folded sums, with implications for open books and Stein fillings in contact topology.

Abstract

It is known that the folded sum of two contact mapping tori whose fibers are compact exact symplectic manifolds having a common convex boundary (called the ``fold'') admits a cooriented contact structure compatible with the obvious fibration map onto the circle. Here we first provide an alternative bundle-theoretical construction of such a ``folded'' contact structure based on a gluing process near the fold. Moreover, we prove that in any odd dimension $2n+1\geq 7$ a folded contact structure on a folded sum of two contact mapping tori is tight if the induced contact form on the (common) contact fold admits no contractible Reeb orbit. In particular, any contact mapping torus of an odd dimension $2n+1\geq 7$ is tight if the induced contact form on the convex boundary of a fiber admits no contractible Reeb orbit.

Tight Contact Structures on Contact Mapping Tori and their Folded Sums

TL;DR

The paper develops a bundle-theoretic framework for folded sums of contact mapping tori and proves the existence of a co-oriented folded-contact structure η compatible with the natural circle fibration, with folded-symplectic on each fiber. In odd dimensions , η is tight provided the induced boundary Reeb dynamics has no contractible periodic orbit, strengthening prior notions of tightness for folded constructions. The authors provide explicit bundle-based constructions via and a gluing-based folded sum approach, illustrating with the folded sum of standard bundles and other Weinstein/Liouville settings. These results broaden the toolkit for constructing tight contact structures on complex fibrations and folded sums, with implications for open books and Stein fillings in contact topology.

Abstract

It is known that the folded sum of two contact mapping tori whose fibers are compact exact symplectic manifolds having a common convex boundary (called the ``fold'') admits a cooriented contact structure compatible with the obvious fibration map onto the circle. Here we first provide an alternative bundle-theoretical construction of such a ``folded'' contact structure based on a gluing process near the fold. Moreover, we prove that in any odd dimension a folded contact structure on a folded sum of two contact mapping tori is tight if the induced contact form on the (common) contact fold admits no contractible Reeb orbit. In particular, any contact mapping torus of an odd dimension is tight if the induced contact form on the convex boundary of a fiber admits no contractible Reeb orbit.
Paper Structure (4 sections, 10 theorems, 39 equations, 2 figures)

This paper contains 4 sections, 10 theorems, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Bundle Contact Structures on Contact Mapping Tori
  3. Folded Contact Structures on Folded Sums of Mapping Tori
  4. Tight Contact Structures on Folded Sums of Mapping Tori

Key Result

Theorem 1.1

For $n\geq 1$, let $\pi_i: \mathcal{M}(\Sigma^{2n}_i,\phi_i) \longrightarrow S^1$ be any two contact mapping tori with the structure groups $\emph{Exact}(\Sigma_i,\partial \Sigma_i, d\beta_i)$ such that $(\Sigma^{2n}_1, d\beta_1)$ and $(\Sigma^{2n}_2, d\beta_2)$ have the same convex boundary. Then t

Figures (2)

  • Figure 1: Smooth functions on the interval $(-1-\epsilon,1+\epsilon)$.
  • Figure 2: Smooth function $h(t)$ on the interval $[-1,1])$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1: Ari
  • Theorem 2.2
  • proof
  • Definition 2.3: Ari
  • Remark 2.4
  • Definition 3.1
  • Definition 3.2
  • ...and 15 more