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.
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Notes
An alternative, less restrictive, assumption on the smooth map \(\pi : M^d \rightarrow \mathbb {R}^k\), is that all their preimages be smoothly stratified spaces of dimension less than or equal to \(\max (0,d-k)\). This restriction has the advantage of being an easily verified hypothesis in examples. The exact definition of the appropriate class of maps has no effect on either of our two theorems.
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This work was partially funded by the "Microsoft Research", "Aspen Center for Physics" and "UCSB".
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Freedman, M. The 2-Width of Embedded 3-Manifolds. Peking Math J 5, 21–35 (2022). https://doi.org/10.1007/s42543-021-00035-9
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DOI: https://doi.org/10.1007/s42543-021-00035-9