The 2-Width of Embedded 3-Manifolds

Abstract

We discuss a possible definition for “k-width” of 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) \rightarrow \infty $. We explain how the divergence of 2-width of embeddings offers a tool to which might prove the Goeritz groups $G_g$ infinitely generated for $g \ge 4$. Finally we construct a homomorphism $\theta _g: G_g \rightarrow \mathrm {MCG}(\underset{g}{\#} S^2 \times S^2)$, suggesting a potential application of 2-width to 4D mapping class groups.

Funding

Microsoft Research

Aspen Center for Physics

UCSB