# The 2-width of embedded 3-manifolds

## TL;DR

This paper introduces a generalized concept of 2-width for 3-manifolds and their embeddings, showing bounds and divergence properties, and explores implications for 4D mapping class groups and Goeritz groups.

## Contribution

It defines 2-width for embedded 3-manifolds, demonstrates bounds and divergence in embeddings, and links these concepts to properties of 4D mapping class groups and Goeritz groups.

## Key findings

- Every closed 3-manifold has 2-width ≤ 2.
- Certain embeddings of T^3 into R^4 have unbounded 2-width.
- Potential to prove infinitely generated Goeritz groups for genus ≥ 4.

## Abstract

We discuss a possible definition for "$k$-width" of both a closed $d$-manifold $M^d$, and on embedding $M^d \overset{e}{\hookrightarrow} \mathbb{R}^n$, $n > d \ge k$, generalizing the classical notion of width of a knot. We show that for every 3-manifold 2-width$(M^3) \le 2$ but that there are embeddings $e_i: T^3 \hookrightarrow \mathbb{R}^4$ with 2-width$(e_i) \to \infty$. We explain how the divergence of 2-width of embeddings offer a tool to which might prove the Goeritz groups $G_g$ infinitely generated for $g \geq 4$. Finally we construct a homeomorphism $\theta_g: G_g \to \mathrm{MCG}(\underset{g}{\#} S^2 \times S^2)$, suggesting a potential application of 2-width to 4D mapping class groups.