Forbidden configurations and definite fillings of lens spaces

TL;DR

This paper classifies lens spaces based on their negative-definite fillings, introducing forbidden configurations that determine the structure of these fillings and have implications for symplectic geometry and singularity theory.

Contribution

It introduces a finite set of forbidden configurations characterizing certain negative-definite fillings of lens spaces and provides a combinatorial framework for their classification.

Findings

Identified 17 forbidden configurations for lens space fillings.

Established a combinatorial framework linking lattice embeddings and plumbing graphs.

Discussed implications for symplectic fillings and cyclic quotient singularities.

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.