Symplectic Fillings of Lens Spaces

• The paper provides a comprehensive classification of minimal symplectic fillings of lens spaces using continued fraction expansions and plumbing graph techniques.
• It employs rational blowdown theory and Lefschetz fibration algorithms to establish precise constraints on Betti numbers and fundamental groups.
• Significant connections are drawn between symplectic fillings, Milnor fibers, and cyclic quotient singularities, revealing deep combinatorial structures.

A symplectic filling of a lens space is a compact symplectic 4-manifold whose boundary is the given lens space equipped with a tight contact structure induced by complex tangencies or via Legendrian surgery. The classification, topology, and combinatorial aspects of symplectic fillings of lens spaces are governed by continued fraction expansions, plumbing graphs, rational blowdown theory, and lattice embedding obstructions. Rigorous bounds relate the fundamental group and second Betti number of the filling, with sharp criteria arising from continued-fraction combinatorics and deep connections to cyclic quotient singularities, Milnor fibers, and the role of rational homology balls, notably with an appearance of Fibonacci numbers in topological constraints.

1. Foundational Definitions and Contact Structures

Let $L(p,q)$ denote the lens space associated to coprime integers $p > q > 0$, defined either as the result of $-p/q$ Dehn surgery on the unknot in $S^3$ or equivalently as the quotient of $S^3 \subset \mathbb{C}^2$ by the free $\mathbb{Z}/p\mathbb{Z}$ action: $$(z_1, z_2) \mapsto \left(e^{2\pi i/p}z_1,\; e^{2\pi i q/p}z_2\right).$$ A contact structure $\xi$ is tight if it contains no overtwisted disks; universally tightness means its lift to the universal cover remains tight, while virtually overtwisted structures arise by Legendrian surgery involving knots stabilized both positively and negatively.

Symplectic fillings $(X, \omega)$ are compact 4-manifolds with boundary $\partial X = L(p,q)$ whose symplectic structure induces a compatible contact structure $\xi$ on the boundary. Minimal fillings are characterized by the absence of symplectic $(-1)$-spheres.

2. Classification via Plumbing and Rational Blowdown

Every minimal symplectic filling of the canonical (Milnor-fillable, universally tight) contact structure on $L(p,q)$ is negative-definite and can be constructed from a linear chain plumbing graph dictated by the Hirzebruch-Jung continued fraction expansion: $$\frac{p}{q} = [a_1, a_2, \ldots, a_\ell],\quad a_i \ge 2,$$ corresponding to spheres with self-intersections $-a_i$. The dual graph $[b_1, \ldots, b_k]$ of $p/(p-q)$ similarly controls the filling. Lisca’s classification asserts that the set of admissible $n$-tuples $(n_1, \ldots, n_k)$—encoding which spheres are present or have been rationally blown down—completely parameterizes all minimal symplectic (Stein) fillings (Etnyre et al., 2020, Bhupal et al., 2013).

Rational blowdown operations replace chains of $(-2)$-spheres with rational homology balls, governed by specific lantern or daisy relations in the planar mapping class group. The Bhupal–Ozbagci rational-blowdown graph organizes all minimal fillings as vertices corresponding to triangulations, with edges representing a single rational blowdown (Bhupal et al., 2022, Bhupal et al., 2013). The graph is graded by Betti number and encodes the possible sequences of rational blowdowns required to pass from the minimal resolution to other fillings.

3. Topological Invariants: Betti Numbers and Fundamental Groups

For a minimal filling $X$ of $(L(p,q), \xi)$, the second Betti number and fundamental group satisfy sharp constraints:

• $d^2$-bound: If $|\pi_1(X)| = d$, then $d^2$ divides $p$ and $b_2(X) \le p/d^2 - 1$, with strict inequality for virtually overtwisted structures (Aceto et al., 2020).
• Fibonacci-bound: For continued fraction $p/q = [a_1,\ldots,a_k]$; if $|\pi_1(X)| \ge F_{\ell+2}$ ($F_n$ = Fibonacci numbers), then $b_2(X) \le k - \ell$. Sharpness occurs exactly for lens spaces of the form $L(n F_{\ell+2}^2,\, n F_{\ell} F_{\ell+2} - 1)$ with universally tight $\xi$.

The maximal $b_2$ occurs for the plumbing graph itself and yields a simply connected filling unless $p$ is not square-free (Fossati, 2019). For virtually overtwisted structures, additional obstructions arise: Stein rational homology ball fillings are impossible except for canonical structures, as detected via Gompf's $d_3$-invariant or lattice theory (Etnyre et al., 2024, Fossati, 2019).

4. Bijection with Milnor Fibers and Algebraic Singularities

Every symplectic filling of $L(p,q)$ (canonical structure) is diffeomorphic to a Milnor fiber smoothing of the cyclic quotient singularity $\frac{1}{p}(1,q)$ (Bhupal et al., 2022, Bhupal et al., 2013). There is a precise combinatorial correspondence:

• Wiring diagrams encode the Lefschetz fibration structure and the planar open book (disk with $k$ holes, monodromy via Dehn twists).
• Each filling corresponds to an incidence matrix matching the smoothing component of the singularity.

For lens spaces $L(p^2, p q - 1)$, the unique rational homology ball filling is recovered as a Milnor fiber and described via explicit Legendrian surgery on appropriate torus knots (Etnyre et al., 2024).

5. Combinatorial Constraints and Forbidden Configurations

Recent lattice embedding results classify when a filling can lower $b_2$ by one below plumbing. Fung–Park (Fung et al., 16 Jan 2026) identify 17 forbidden linear subgraph configurations in the canonical plumbing whose absence ensures that the only possible fillings are the plumbing and its single rational blowdown along designated $-4$ sphere, $(-3)$-$(-3)$, or $(-3)$-$(-2)$-$(-3)$. Minimal fillings for lens spaces realize the lowest possible $b_2$, with possible nontrivial fundamental group only for $L(4,1)$, $L(8,3)$, $L(12,5)$.

6. Explicit Construction: Lefschetz Fibrations and Wiring Diagrams

Planar Lefschetz fibrations associated to fillings are constructed via combinatorial algorithms:

• Blowup-twisting algorithm builds wiring diagrams whose monodromy matches the Dehn twist factorization for the filling (Bhupal et al., 2022).
• Each filling yields a planar arrangement of symplectic graphical disks in $\mathbb{C}^2$; blowing up intersection points and removing proper transforms recovers the Stein filling and its Lefschetz structure.

This framework incorporates all classical fillings as special cases, with each planar diagram admitting a bijection to a Milnor fiber.

7. Examples and Classification Criteria

• Hopf link surgeries: Length-2 lens spaces with $a_i \neq 4$ have unique filling; $a_j=4$ (rotation number $\pm2$) introduce a second filling via lantern relation (Fossati, 2019, Fossati, 2020).
• Canonical rational balls: Only $L(p^2, p q - 1)$ with canonical contact structure admit rational homology ball fillings (Etnyre et al., 2024).
• Symplectic cobordisms: Maps between fillings of lens spaces require nondecreasing continued fraction length; subchain cobordisms and rolled-up diagrams explicate structural transitions (Etnyre et al., 2020).

8. Open Problems and Future Directions

Key open questions:

• Extension of precise $\pi_1$ vs.\ $b_2$ trade-off bounds to broader classes such as general Seifert fibered spaces.
• Investigation of the impact of these bounds on the geography of symplectic 4-manifolds with rational homology sphere boundary.
• Characterization of virtually overtwisted structures optimizing Betti number gaps.

These results unify and generalize previously scattered results, highlighting new quantitative invariants and the role of combinatorics in 4-dimensional symplectic topology (Aceto et al., 2020).

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References:

• (Aceto et al., 2020, Bhupal et al., 2022, Fossati, 2019, Fossati, 2019, Etnyre et al., 2024, Fossati, 2020, Etnyre et al., 2020, Christian et al., 2020, Bhupal et al., 2022, Fung et al., 16 Jan 2026, Bhupal et al., 2013)