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Simplifying 3-manifolds in 4
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 5, pp. 1079-1101.

Nous montrons qu’un plongement lisse d’une 3-variété fermée dans S 3 × peut être isotopé sorte que chaque niveau générique divise S 3 ×t en deux corps d’anses (i.e., est Heegaard) à condition que le plongement original a un maximum local unique par rapport à la coordonnée . Ceci permet d’étudier l’unicité des plongements en utilisant le groupe de classe de difféotopies des surfaces et la conjecture de Schoenflies est considérée dans cette optique. Nous donnons aussi une condition nécessaire et suffisante pour que la somme connexe d’une 3-variété et d’un nombre arbitraire de copies de S 1 ×S 2 se plonge dans 4 .

We show that a smooth embedding of a closed 3-manifold in S 3 × can be isotoped so that every generic level divides S 3 ×t into two handlebodies (i.e., is Heegaard) provided the original embedding has a unique local maximum with respect to the coordinate. This allows uniqueness of embeddings to be studied via the mapping class group of surfaces and the Schoenflies conjecture is considered in this light. We also give a necessary and sufficient condition that a 3-manifold connected summed with arbitrarily many copies of S 1 ×S 2 embeds in 4 .

Publié le :
DOI : https://doi.org/10.5802/afst.1476
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     author = {Ian Agol and Michael Freedman},
     title = {Simplifying $3$-manifolds in ${\mathbb{R}}^4$},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1079--1101},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 24},
     number = {5},
     year = {2015},
     doi = {10.5802/afst.1476},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1476/}
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Ian Agol; Michael Freedman. Simplifying $3$-manifolds in ${\mathbb{R}}^4$. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 5, pp. 1079-1101. doi : 10.5802/afst.1476. https://afst.centre-mersenne.org/articles/10.5802/afst.1476/

[1] Akbulut (S.) and Kirby (R.).— A potential smooth counterexample in dimension 4 to the Poincaré conjecture, the Schoenflies conjecture, and the Andrews-Curtis conjecture, Topology 24, no. 4, p. 375-390 (1985). | MR 816520 | Zbl 0584.57009

[2] Budney (R.) and Burton (B. A.).— Embeddings of 3-manifolds in S 4 from the point of view of the 11-tetrahedron census, ArXiv e-prints (2008), 49 pages, 0810.2346.

[3] Cerf (J.).— Sur les difféomorphismes de la sphère de dimension trois (Γ 4 =0), Lecture Notes in Mathematics, No. 53, Springer-Verlag, Berlin (1968). | MR 229250 | Zbl 0164.24502

[4] Frohman (C.).— An unknotting lemma for systems of arcs in F×I, Pacific J. Math. 139, no. 1, p. 59-66 (1989). | MR 1010785 | Zbl 0693.57010

[5] Goeritz (L.).— Die abbildungen der brezelfläche und der volbrezel vom gesschlect 2, Abh. Math. Sem. Univ. Hamburg 9, p. 244-259 (1933). | MR 3069602 | Zbl 0007.08102

[6] Gompf (R. E.).— Killing the Akbulut-Kirby 4-sphere, with relevance to the Andrews-Curtis and Schoenflies problems, Topology 30, no. 1, p. 97-115 (1991). | MR 1081936 | Zbl 0715.57016

[7] Hatcher (A.) and Wagoner (J.).— Pseudo-isotopies of compact manifolds, Asterisque, no. 6. Soc. Math. de France, Paris (1973). | MR 353337 | Zbl 0274.57010

[8] Jaco (W.).— Adding a 2-handle to a 3-manifold: an application to property R, Proc. Amer. Math. Soc. 92, no. 2, p. 288-292 (1984). | MR 754723 | Zbl 0564.57009

[9] Levine (J.).— Some results on higher dimensional knot groups, Knot theory (Proc. Sem., Plans-sur-Bex, 1977), Lecture Notes in Math., vol. 685, Springer, Berlin (1978), With an appendix by Claude Weber, p. 243-273. | MR 521737 | Zbl 0404.57019

[10] Livingston (C.).— Four-manifolds of large negative deficiency, Math. Proc. Cambridge Philos. Soc. 138, no. 1, p. 107-115 (2005). | MR 2127231 | Zbl 1079.57021

[11] Massey (W. S.).— Proof of a conjecture of Whitney, Pacific J. Math.  31, p. 143-156 (1969). | MR 250331 | Zbl 0198.56701

[12] Scharlemann (M.).— The four-dimensional Schoenflies conjecture is true for genus two imbeddings, Topology 23, no. 2, p. 211-217 (1984). | MR 744851 | Zbl 0543.57011

[13] Scharlemann (M.).— Generalized Property R and the Schoenflies Conjecture, Commentarii Mathematici Helvetici 83, p. 421-449 (2008). | MR 2390052 | Zbl 1148.57032

[14] Scharlemann (M.) Smooth spheres in R 4 with four critical points are standard, Invent. Math. 79, no. 1, p. 125-141 (1985). | MR 774532 | Zbl 0559.57019

[15] Trace (B.).— On attaching 3-handles to a 1-connected 4-manifold, Pacific J. Math. 99, no. 1, p. 175-181 (1982). | MR 651494 | Zbl 0531.57010

[16] Waldhausen (F.).— Heegaard-Zerlegungen der 3-Sphäre, Topology 7, p. 195-203 (1968). | MR 227992 | Zbl 0157.54501

[17] Zeeman (E. C.).— On the dunce hat, Topology 2, p. 341-358 (1964). | MR 156351 | Zbl 0116.40801

[18] Zeeman (E. C.) Twisting spun knots, Trans. Amer. Math. Soc. 115, p. 471-495 (1965). | MR 195085 | Zbl 0134.42902

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