The 2-width of embedded 3-manifolds

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 $g\ge 4$. Finally we construct a homeomorphism $\theta_g: G_g \to $\mathrm{M}\mathrm{C}\mathrm{G}$(\underset{g}{#} S^2 \times S^2)$, suggesting a potential application of 2-width to 4D mapping class groups.