A survey on embeddings of 3-manifolds in definite 4-manifolds

TL;DR

This survey reviews recent advances in embedding 3-manifolds into definite 4-manifolds, highlighting Donaldson's theorem, lattice combinatorics, and introducing new results on amphichiral lens spaces.

Contribution

It provides a comprehensive overview of current techniques and presents a novel result on embedding amphichiral lens spaces in negative-definite 4-manifolds.

Findings

Emphasizes the importance of Donaldson's diagonalization theorem.

Highlights the role of integral lattice combinatorics.

Introduces new embedding results for amphichiral lens spaces.

Abstract

This article presents a survey on the topic of embedding 3-manifolds in definite 4-manifolds, emphasizing the latest progress in the field. We will focus on the significant role played by Donaldson's diagonalization theorem and the combinatorics of integral lattices in understanding these embeddings. Additionally, the article introduces a new result concerning the embedding of amphichiral lens spaces in negative-definite manifolds.