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.
Similar content being viewed by others
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.
References
Asano, K.: A note on surfaces in 4-spheres. Mat. Sem. Notes Kobe Univ. 4(2), 195–198 (1976)
Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Mathematical Series, vol. 49. Princeton University, Princeton, NJ (2011)
Freedman, M.: Group width. Math. Res. Lett. 18(3), 433–436 (2011)
Freedman, M., Scharlemann, M.: Powell moves and the Goeritz group. arXiv:1804.05909v1 (2018)
Freedman, M., Hillman, J.: Width of codimension two knots. J. Knot Theory Ramifications 29, 1950094, 8 pp. (2020)
Gabai, D.: Foliations and the topology of 3-manifolds III. J. Differ. Geom. 26, 479–536 (1987)
Gibson, C., Wirthmüller, K., du Plessis, A., Looijenga, E.: Topological Stability of Smooth Mappings. Lecture Notes in Mathematics, vol. 552. Springer, Berlin (1976)
Ketover, D.: Genus bounds for min-max minimal surfaces. J. Differ. Geom. 112(3), 555–590 (2019)
Litherland, R.A.: Deforming twist-spun knots. Trans. Amer. Math. Soc. 250, 311–331 (1979)
Montesinos, J.: A note on 3-fold branched coverings of S3. Math. Proc. Camb. Philos. Soc. 88(2), 321–325 (1980)
Piergallini, R.: Four-manifolds as 4-fold branched covers of S4. Topology 34(3), 497–508 (1995)
Pitts, J. Rubenstein, J. H.: Equivariant minimax and minimal surfaces in geometric three-manifolds. Bull. Amer. Math. Soc. (N.S.) 19(1), 303–309 (1988)
Powell, J.: Homeomorphisms of S3 leaving a Heegaard surface invariant. Trans. Amer. Math. Soc. 257(1), 193–216 (1980)
Scharlemann, M.: One Powell generator is redundant. Proc. Amer. Math. Soc. Ser. B 7, 138–141 (2020)
Scharlemann, M., Thompson, A.: Thin position for 3-manifolds. In: Geometric Topology (Haifa, 1992), 231–238, Contemp. Math., vol. 164, Amer. Math. Soc. Providence, RI (1994)
Thompson, A.: Thin position and the recognition problem for S3. Math. Res. Lett. 1, 613–630 (1994)
Funding
This work was partially funded by the "Microsoft Research", "Aspen Center for Physics" and "UCSB".
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42543-021-00035-9