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Forbidden configurations and definite fillings of lens spaces

Antony T. H. Fung, JungHwan Park

TL;DR

This work addresses the problem of classifying definite (negative-definite) fillings of lens spaces by introducing a dual-plumbing lattice framework and a finite obstruction set S_k of forbidden configurations. The authors prove that, for each k, a connected sum L = ∑ L(p_i,q_i) satisfies a lattice-embedding property X_k if and only if the dual plumbing avoids S_k, yielding lower bounds on b_2 and a complete description of intersection forms up to ⟨−1⟩-summands; for k=1 and k=2 they obtain explicit finite sets (|S_1|=10 and |S_2|=17) and fully classify negative-definite fillings. The results connect to symplectic fillings and Milnor fibers via Lisca’s classification and rational blowdowns, providing a practical finite obstruction framework to determine how lens spaces bound 4-manifolds. This advances the understanding of which 4-manifolds bound a given lens space and demonstrates the role of combinatorial graph configurations in encoding lattice-embedding obstructions.

Abstract

We study definite fillings of lens spaces. We classify the lens spaces $L(p,q)$ for which every smooth negative-definite filling $X$ satisfies \[ b_2(X)\ge b_2(X(p,q))-1, \] where $X(p,q)$ denotes the canonical negative-definite plumbing. The classification is given by 17 "forbidden configurations" that cannot appear as induced subgraphs of the canonical plumbing graph. More generally, we introduce a combinatorial framework that encodes the lattice embedding information coming from the dual plumbing of $X(p,q)$, and we prove that it is governed by a finite set of minimal forbidden configurations. We also discuss consequences for symplectic fillings of lens spaces and for smoothings of cyclic quotient singularities.

Forbidden configurations and definite fillings of lens spaces

TL;DR

This work addresses the problem of classifying definite (negative-definite) fillings of lens spaces by introducing a dual-plumbing lattice framework and a finite obstruction set S_k of forbidden configurations. The authors prove that, for each k, a connected sum L = ∑ L(p_i,q_i) satisfies a lattice-embedding property X_k if and only if the dual plumbing avoids S_k, yielding lower bounds on b_2 and a complete description of intersection forms up to ⟨−1⟩-summands; for k=1 and k=2 they obtain explicit finite sets (|S_1|=10 and |S_2|=17) and fully classify negative-definite fillings. The results connect to symplectic fillings and Milnor fibers via Lisca’s classification and rational blowdowns, providing a practical finite obstruction framework to determine how lens spaces bound 4-manifolds. This advances the understanding of which 4-manifolds bound a given lens space and demonstrates the role of combinatorial graph configurations in encoding lattice-embedding obstructions.

Abstract

We study definite fillings of lens spaces. We classify the lens spaces for which every smooth negative-definite filling satisfies where denotes the canonical negative-definite plumbing. The classification is given by 17 "forbidden configurations" that cannot appear as induced subgraphs of the canonical plumbing graph. More generally, we introduce a combinatorial framework that encodes the lattice embedding information coming from the dual plumbing of , and we prove that it is governed by a finite set of minimal forbidden configurations. We also discuss consequences for symplectic fillings of lens spaces and for smoothings of cyclic quotient singularities.
Paper Structure (8 sections, 29 theorems, 114 equations, 4 figures)

This paper contains 8 sections, 29 theorems, 114 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Rational homology balls and rational blowdowns
  3. Working Conditions for dual plumbing graphs
  4. Standard and semi-standard embeddings
  5. Theorem \ref{['thm:main']} from the Working Conditions
  6. Forbidden configurations
  7. Symplectic fillings and Milnor fibers
  8. Computer data

Key Result

Lemma 1.2

∎ Let $L = \#_i L(p_i, q_i)$ be a connected sum of lens spaces, and let $\natural_i X(p_i, q_i)$ denote the corresponding boundary connected sum of the canonical negative-definite plumbings. If $L$ satisfies Property $X_k$ with a positive integer $k$, then every smooth negative-definite filling $X$

Figures (4)

  • Figure 1: Rational blowdown.
  • Figure 2: Blowdown for $[-4]$.
  • Figure 3: Blowdown for $[-3,-3]$.
  • Figure 4: Handle slide for the $[-3,-2,-3]$ case.

Theorems & Definitions (73)

  • Definition 1.1
  • Lemma 1.2
  • Theorem A
  • Theorem B
  • Corollary 1.3
  • Remark 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Remark 1.7
  • Lemma 2.1
  • ...and 63 more