[Paper Review] On symplectic 4-manifolds with prescribed fundamental group

Scott Baldridge, Paul Kirk|ArXiv.org|Apr 17, 2005

Geometric and Algebraic Topology20 references4 citations

TL;DR

This paper investigates the minimization of topological invariants—specifically the Euler characteristic $\chi$ and $\chi + \sigma$—among closed symplectic 4-manifolds with a prescribed fundamental group $G$. Using group presentations, the authors construct explicit symplectic 4-manifolds achieving upper bounds on $\chi$ and $\chi + \sigma$, and prove that for $G = \mathbb{Z}^{2g}$ with $g \equiv 0,1,3 \pmod{4}$, the symmetric square $S_g = \mathrm{Sym}^2(F_g)$ minimizes $\chi$ among all such manifolds.

ABSTRACT

In this article we study the problem of minimizing $aχ+bσ$ on the class of all symplectic 4--manifolds with prescribed fundamental group $G$ ($χ$ is the Euler characteristic, $σ$ is the signature, and $a,b\in \BR$), focusing on the important cases $χ$, $χ+σ$ and $2χ+3σ$. In certain situations we can derive lower bounds for these functions and describe symplectic 4-manifolds which are minimizers. We derive an upper bound for the minimum of $χ$ and $χ+σ$ in terms of the presentation of $G$.

Motivation & Objective

• To determine the minimal possible values of $\chi$ and $\chi + \sigma$ among closed symplectic 4-manifolds with a given fundamental group $G$.
• To derive upper bounds for $\min \chi(M)$ and $\min (\chi + \sigma)(M)$ in terms of the number of generators and relations in a presentation of $G$.
• To identify explicit constructions of symplectic 4-manifolds that achieve these minimal values for specific groups, particularly free abelian groups.
• To explore the uniqueness of minimizers up to diffeomorphism and investigate conditions under which minimizers are uniquely determined by their intersection form.

Proposed method

• Constructing symplectic 4-manifolds via double of 2-handlebodies from group presentations, ensuring $\pi_1(M) \cong G$.
• Applying Gompf's method to produce symplectic structures on manifolds with prescribed fundamental groups.
• Using the deficiency of a group presentation to bound the Euler characteristic from above: $\min \chi(M) \leq 2 - 2\text{def}(G)$.
• Establishing that for $G = \mathbb{Z}^{2g}$ with $g \equiv 0,1,3 \pmod{4}$, the symmetric square $S_g = \mathrm{Sym}^2(F_g)$ is a minimizer of $\chi$.
• Combining standard bounds for 4-manifolds with symplectic-specific constraints: $b^+ \geq 1$ and $1 - b_1 + b^+$ even, to derive lower bounds on $\chi(M)$.
• Speculating on deeper symplectic invariants to improve lower bounds, particularly by leveraging the symplectic structure beyond basic Betti number constraints.

Experimental results

Research questions

• RQ1 What is the minimal value of the Euler characteristic $\chi$ among all closed symplectic 4-manifolds with fundamental group $G$?
• RQ2 Can the minimal value of $\chi + \sigma$ be achieved, and are such minimizers unique up to diffeomorphism?
• RQ3 How do the number of generators and relations in a presentation of $G$ influence the upper bounds on $\min \chi(M)$ and $\min (\chi + \sigma)(M)$?
• RQ4 For which groups $G$ does the symmetric square $\mathrm{Sym}^2(F_g)$ of a genus-$g$ surface realize the minimal $\chi$ among symplectic 4-manifolds with $\pi_1 = \mathbb{Z}^{2g}$?
• RQ5 Are there deeper symplectic invariants that can improve the lower bounds on $\min \chi(M)$ beyond those derived from Betti number constraints?

Key findings

• For any finitely presented group $G$ with $g$ generators and $r$ relations, there exists a closed symplectic 4-manifold $M$ with $\pi_1(M) \cong G$, $\chi(M) = 12(g + r + 1)$, and $\sigma(M) = -8(g + r + 1)$, establishing an explicit upper bound for $\chi$ and $\sigma$.
• When $G = \mathbb{Z}^{2g}$ and $g \equiv 0,1,3 \pmod{4}$, the symmetric square $S_g = \mathrm{Sym}^2(F_g)$ minimizes $\chi$ among all symplectic 4-manifolds with fundamental group $\mathbb{Z}^{2g}$.
• The difference between the minimal $\chi$ for symplectic manifolds and for smooth 4-manifolds with $\pi_1 = \mathbb{Z}^{2g}$ is at most one, indicating tight control in this case.
• For $G = \mathbb{Z}^{2g}$, the minimal $\chi$ is realized by $S_g$ only when $g \equiv 0,1,3 \pmod{4}$, suggesting a number-theoretic obstruction to minimality.
• The authors conjecture that minimizers of $\chi$ are uniquely determined up to diffeomorphism by their intersection form $Q_M$, suggesting a strong link between topology and minimality.
• A key open problem is to find a symplectic 4-manifold $K$ with $\chi(K) < 12$ containing a symplectically embedded torus of self-intersection zero with $\pi_1(K - T) \cong \mathbb{Z}$ or trivial, as such a manifold would improve the upper bound construction.