The 2-width of embedded 3-manifolds

Abstract

We discuss a possible definition for "$k$-width" of both a closed $d$-manifold $M^d$, and on embedding $M^d\stackrel{e}{\hookrightarrow }{\mathbb{R}}^n$, $n>d\ge k$, 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 ${\theta }_g:G_g\to \mathrm{MCG}(\underset{g}{\#}{S}^2\times S^2)$, suggesting a potential application of 2-width to 4D mapping class groups.

References in corpus (1)

• Width of codimension two knots