  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.
