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Surgery on tori in the 4–sphere

Published online by Cambridge University Press:  27 September 2016

KYLE LARSON
KYLE LARSON*
Affiliation:
Michigan State University e-mail: [email protected]
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Abstract

We investigate the operation of torus surgery on tori embedded in S4. Key questions include which 4–manifolds can be obtained in this way, and the uniqueness of such descriptions. As an application we construct embeddings of 3–manifolds into 4–manifolds by viewing Dehn surgery as a cross section of a surgery on a surface. In particular, we give new embeddings of homology spheres into S4.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 1 , January 2018 , pp. 109 - 124
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Asimov, D. Round handles and non-singular Morse–Smale flows. Ann. of Math. (2), 102 (1975), pp. 41–54.Google Scholar
[2] Baykur, R. İ. and Sunukjian, N. Round handles, logarithmic transforms and smooth 4-manifolds. J. Topol. 6 (2013), pp. 49–63.CrossRefGoogle Scholar
[3] Boyle, J. The turned torus knot in S 4 . J. Knot Theory Ramifications. 2 (1993), pp. 239–249.Google Scholar
[4] Budney, R. and Burton, B. A. Embeddings of 3-manifolds in S 4 from the point of view of the 11-tetrahedron census. Preprint available at http://arxiv.org/abs/0810.2346, 2012.Google Scholar
[5] Carter, S., Kamada, S. and Saito, M. Surfaces in 4-space. Encyclopaedia of Mathematical Sciences, vol. 142 (Springer-Verlag, Berlin, 2004). Low-Dimensional Topology, III.Google Scholar
[6] Crisp, J. S. and Hillman, J. A. Embedding Seifert fibred 3-manifolds and Sol3-manifolds in 4-space. Proc. London Math. Soc. (3), 76 (1998), pp. 685–710.Google Scholar
[7] Donald, A. Embedding Seifert manifolds in S 4 . Trans. Amer. Math. Soc. 367 (2015), pp. 559–595.Google Scholar
[8] Edmonds, A. L. and Livingston, C. Embedding punctured lens spaces in four-manifolds. Comment. Math. Helv. 71 (1996), pp. 169–191.CrossRefGoogle Scholar
[9] Fintushel, R. and Stern, R. J. Six lectures on four 4-manifolds. in Low dimensional topology, IAS/Park City Math. Ser., vol. 15, (Amer. Math. Soc., Providence, RI, 2009), pp. 265–315.Google Scholar
[10] Freedman, M. H. The topology of four-dimensional manifolds. J. Differential Geom. 17 (1982), pp. 357–453.CrossRefGoogle Scholar
[11] Gilmer, P. M. and Livingston, C. On embedding 3-manifolds in 4-space. Topology 22 (1983), pp. 241–252.CrossRefGoogle Scholar
[12] Gluck, H. The embedding of two-spheres in the four-sphere. Trans. Amer. Math. Soc. 104 (1962), pp. 308–333.CrossRefGoogle Scholar
[13] Gompf, R. E. More Cappell-Shaneson spheres are standard. Algebr. Geom. Topol. 10 (2010), pp. 1665–1681.Google Scholar
[14] Gompf, R. E. Rough notes on the Cappell–Shaneson construction (2012).Google Scholar
[15] Gompf, R. E. and Stipsicz, A. I. 4-manifolds and Kirby calculus. Graduate Studies in Math., vol. 20 (American Mathematical Society, Providence, RI, 1999).Google Scholar
[16] Gordon, C. M. Knots in the 4-sphere. Comment. Math. Helv. 51 (1976), pp. 585–596.Google Scholar
[17] Gordon, C. M. and Luecke, J. Knots are determined by their complements. J. Amer. Math. Soc. 2 (1989), pp. 371–415.Google Scholar
[18] Hausmann, J.-C. and Weinberger, S.. Caractéristiques d'Euler et groupes fondamentaux des variétés de dimension 4. Comment. Math. Helv. 60 (1985), pp. 139–144.Google Scholar
[19] Hirose, S. On diffeomorphisms over T 2-knots. Proc. Amer. Math. Soc. 119 (1993), pp. 1009–1018.Google Scholar
[20] Iwase, Z. Good torus fibrations with twin singular fibers. Japan. J. Math. (N.S.) 10 (1984), pp. 321–352.Google Scholar
[21] Iwase, Z. Dehn-surgery along a torus T 2-knot. Pacific J. Math. 133 (1988), pp. 289–299.Google Scholar
[22] Iwase, Z. Dehn surgery along a torus T 2-knot. II. Japan. J. Math. (N.S.) 16 (1990), pp. 171–196.Google Scholar
[23] Kervaire, M. A. Smooth homology spheres and their fundamental groups. Trans. Amer. Math. Soc. 144 (1969), pp. 67–72.Google Scholar
[24] Larson, K. and Meier, J. Fibered ribbon disks. J. Knot Theory Ramifications. 24 (2015), p. 1550066.Google Scholar
[25] Laudenbach, F. and Poénaru, V. A note on 4-dimensional handlebodies. Bull. Soc. Math. France. 100 (1972), pp. 337–344.CrossRefGoogle Scholar
[26] Levine, J. Some results on higher dimensional knot groups. In Knot Theory (Proc. Sem., Plans-sur-Bex, 1977) Lecture Notes in Math., vol. 685 (Springer, Berlin, 1978), pp. 243–273. With an appendix by Claude Weber.Google Scholar
[27] Livingston, C. Stably irreducible surfaces in S 4 . Pacific J. Math. 116 (1985), pp. 77–84.Google Scholar
[28] Meier, J. Distinguishing topologically and smoothly doubly slice knots. J. Topol. 8 (2015), pp. 315–351.Google Scholar
[29] Montesinos, J. M. Heegaard diagrams for closed 4-manifolds. In Geometric Topology (Proc. Georgia Topology Conf., Athens, Ga., 1977) (Academic Press, New York-London, 1979), pp. 219–237.CrossRefGoogle Scholar
[30] Montesinos, J. M. On twins in the four-sphere. I. Quart. J. Math. Oxford Ser. (2), 34 (1983), pp. 171–199.Google Scholar
[31] Pao, P. S. The topological structure of 4-manifolds with effective torus actions. I. Trans. Amer. Math. Soc. 227 (1977), pp. 279–317.Google Scholar
[32] Rokhlin, V. A. Proof of Gudkov's hypothesis. Funkcional. Anal. i Priložen. 6 (1972), pp. 62–64.Google Scholar