[Paper Review] Positive Dehn Twist Expressions for Some Elements of Finite Order in the Mapping Class Group

Yusuf Z. Gürtaş|ArXiv.org|Jan 22, 2005

Geometric and Algebraic Topology2 references3 citations

TL;DR

This paper presents explicit positive Dehn twist expressions for finite-order mapping class group elements that act as $2\pi/p$ rotations on genus-$g$ surfaces, using compositions of involutions derived from prior work. It computes the homeomorphism invariants—Euler characteristic $\chi = 2p(p+7)$ and signature $\sigma = -12p$—of the resulting simply connected symplectic 4-manifolds realized as Lefschetz fibrations.

ABSTRACT

Positive Dehn twist products for some elements of finite order in the mapping class group of a 2-dimensional closed, compact, oriented surface $Σ_g$, which are rotations of $Σ_g$ through $2π/p$, are presented. The homeomorphism invariants of the resulting simply connected symplectic 4- manifolds are computed.

Motivation & Objective

• To construct explicit positive Dehn twist expressions for finite-order mapping class group elements that act as rotational symmetries on closed, oriented surfaces of genus $g = 1+p$.
• To extend prior results on involutions in the mapping class group to higher-order rotational symmetries using composition of $180^\circ$ involutions.
• To compute the homeomorphism invariants—Euler characteristic and signature—of the symplectic 4-manifolds obtained as Lefschetz fibrations over $S^2$ with monodromy given by these rotational elements.
• To establish a closed-form formula for the signature of these fibrations, supported by computational evidence from a MATLAB implementation.
• To demonstrate that the resulting 4-manifolds are simply connected and admit almost complex structures, enabling computation of $c_1^2$ and $\chi_h$.

Proposed method

• Construct the rotation $\phi_p$ of order $p$ on $\Sigma_{1+p}$ as the composition $\phi_p = \theta_2^p \theta_1^p$, where $\theta_1^p$ and $\theta_2^p$ are $180^\circ$ involutions about specified axes.
• Apply Theorem 1.0.1 to express each involution $\theta_1^p$ and $\theta_2^p$ as a product of positive Dehn twists using labeled cycles on the surface.
• Juxtapose the Dehn twist expressions for $\theta_1^p$ and $\theta_2^p$ to obtain the full positive Dehn twist expression for $\phi_p = \theta_2^p \theta_1^p$.
• Use the Lefschetz fibration formula $\chi(X) = 4 - 4g + s$, where $s = 2p(p+9)$ is the number of vanishing cycles (singular fibers), to compute the Euler characteristic.
• Employ a MATLAB program based on the algorithm in [3] to compute the signature $\sigma(X)$, leading to the closed-form conjecture $\sigma(X) = -12p$.
• Derive $c_1^2(X) = 4p(p-2)$ and $\chi_h(X) = \frac{1}{2}p(p+1)$ using standard formulas: $c_1^2 = 3\sigma + 2\chi$ and $\chi_h = (\sigma + \chi)/4$.

Experimental results

Research questions

• RQ1 Can positive Dehn twist expressions be constructed for finite-order mapping class group elements that act as $2\pi/p$ rotations on genus-$g$ surfaces?
• RQ2 How can such rotational symmetries be decomposed into compositions of known $180^\circ$ involutions with known Dehn twist expressions?
• RQ3 What are the homeomorphism invariants—specifically Euler characteristic and signature—of the symplectic 4-manifolds obtained as Lefschetz fibrations with monodromy $\phi_p$?
• RQ4 Is there a closed-form formula for the signature of these fibrations, and can it be verified computationally?
• RQ5 Do the resulting 4-manifolds admit almost complex structures, and what are their Chern class invariants?

Key findings

• The $2\pi/p$ rotation $\phi_p$ on $\Sigma_{1+p}$ is realized as a composition $\phi_p = \theta_2^p \theta_1^p$, where $\theta_1^p$ and $\theta_2^p$ are $180^\circ$ involutions with known positive Dehn twist expressions.
• The positive Dehn twist expression for $\phi_p$ is obtained by concatenating the Dehn twist sequences for $\theta_1^p$ and $\theta_2^p$, resulting in a product of $2p(p+9)$ Dehn twists.
• The Euler characteristic of the associated Lefschetz fibration is $\chi(X) = 2p(p+7)$, derived from the formula $\chi = 4 - 4g + s$ with $g = p+1$ and $s = 2p(p+9)$.
• The signature of the fibration is computed as $\sigma(X) = -12p$, based on computational results from a MATLAB program using the algorithm in [3].
• The square of the first Chern class is $c_1^2(X) = 4p(p-2)$, and the holomorphic Euler characteristic is $\chi_h(X) = \frac{1}{2}p(p+1)$, confirming the manifold is simply connected and admits an almost complex structure.
• The paper provides explicit outputs of signature computations for $p = 3, 5, 7, 9$, showing consistent patterns that support the closed-form signature formula.