Definite fillings of lens spaces

TL;DR

This paper classifies lens spaces with minimal negative-definite fillings, identifying forbidden plumbing subgraphs and showing uniqueness of intersection forms for minimal fillings, with implications for embedding lens spaces in 4-manifolds.

Contribution

It provides a classification of minimal negative-definite fillings of lens spaces using forbidden subgraphs and proves the uniqueness of their intersection forms.

Findings

Identified 10 forbidden subgraphs for minimal fillings.

Proved minimal fillings have unique intersection forms up to diagonal summands.

Discussed implications for smooth embeddings of lens spaces in 4-manifolds.

Abstract

This paper considers the problem of determining the smallest (as measured by the second Betti number) smooth negative-definite filling of a lens space. The main result is to classify those lens spaces for which the associated negative-definite canonical plumbing is minimal. The classification takes the form of a list of 10 "forbidden" subgraphs that cannot appear in the plumbing graph if the corresponding plumbed 4-manifold is minimal. We also show that whenever the plumbing is minimal any other negative-definite filling for the given lens space has the same intersection form up to addition of diagonal summands. Consequences regarding smooth embeddings of lens spaces in 4-manifolds are also discussed.