Brieskorn spheres bounding rational balls
HTML articles powered by AMS MathViewer
- by Selman Akbulut and Kyle Larson PDF
- Proc. Amer. Math. Soc. 146 (2018), 1817-1824 Request permission
Abstract:
Fintushel and Stern showed that the Brieskorn sphere $\Sigma (2,3,7)$ bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to provide the first additional examples of Brieskorn spheres that bound rational homology balls but not integral homology balls: the families $\Sigma (2,4n+1,12n+5)$ and $\Sigma (3,3n+1,12n+5)$ for $n$ odd. We also provide handlebody diagrams for a rational homology ball containing a rationally slice disk for the figure-eight knot, as well as for a rational homology ball bounded by $\Sigma (2,3,7)$. These handle diagrams necessarily contain 3-handles.References
- Selman Akbulut and Robion Kirby, Mazur manifolds, Michigan Math. J. 26 (1979), no. 3, 259–284. MR 544597
- Selman Akbulut, 4-manifolds, Oxford Graduate Texts in Mathematics, vol. 25, Oxford University Press, Oxford, 2016. MR 3559604, DOI 10.1093/acprof:oso/9780198784869.001.0001
- Paolo Aceto and Kyle Larson, Knot concordance and homology sphere groups, To appear in Int. Math. Res. Not. IMRN, 2016.
- Andrew J. Casson and John L. Harer, Some homology lens spaces which bound rational homology balls, Pacific J. Math. 96 (1981), no. 1, 23–36. MR 634760
- Jae Choon Cha, The structure of the rational concordance group of knots, Mem. Amer. Math. Soc. 189 (2007), no. 885, x+95. MR 2343079, DOI 10.1090/memo/0885
- Henry Clay Fickle, Knots, $\textbf {Z}$-homology $3$-spheres and contractible $4$-manifolds, Houston J. Math. 10 (1984), no. 4, 467–493. MR 774711
- Ronald Fintushel and Ronald J. Stern, An exotic free involution on $S^{4}$, Ann. of Math. (2) 113 (1981), no. 2, 357–365. MR 607896, DOI 10.2307/2006987
- Ronald Fintushel and Ronald J. Stern, A $\mu$-invariant one homology $3$-sphere that bounds an orientable rational ball, Four-manifold theory (Durham, N.H., 1982) Contemp. Math., vol. 35, Amer. Math. Soc., Providence, RI, 1984, pp. 265–268. MR 780582, DOI 10.1090/conm/035/780582
- Akio Kawauchi, The (2,1)-cable of the figure eight knot is rationally slice, handwritten manuscript, 1980.
- Akio Kawauchi, Rational-slice knots via strongly negative-amphicheiral knots, Commun. Math. Res. 25 (2009), no. 2, 177–192. MR 2554510
- Min Hoon Kim and Zhongtao Wu, On rational sliceness of Miyazaki’s fibered, amphicheiral knots, Preprint available at https://arxiv.org/abs/1604.04870, 2016.
- Walter D. Neumann, An invariant of plumbed homology spheres, Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), Lecture Notes in Math., vol. 788, Springer, Berlin, 1980, pp. 125–144. MR 585657
- Walter D. Neumann and Frank Raymond, Seifert manifolds, plumbing, $\mu$-invariant and orientation reversing maps, Algebraic and geometric topology (Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977) Lecture Notes in Math., vol. 664, Springer, Berlin, 1978, pp. 163–196. MR 518415
- Nikolai Saveliev, Invariants for homology $3$-spheres, Encyclopaedia of Mathematical Sciences, vol. 140, Springer-Verlag, Berlin, 2002. Low-Dimensional Topology, I. MR 1941324, DOI 10.1007/978-3-662-04705-7
- Ronald J. Stern, Some Brieskorn spheres which bound contractible manifolds, Notices Amer. Math. Soc 25 (1978), A448.
Additional Information
- Selman Akbulut
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48823
- MR Author ID: 23925
- Email: akbulut@math.msu.edu
- Kyle Larson
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48823
- MR Author ID: 1090744
- Email: larson@math.msu.edu
- Received by editor(s): April 27, 2017
- Received by editor(s) in revised form: May 16, 2017
- Published electronically: October 30, 2017
- Communicated by: David Futer
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 1817-1824
- MSC (2010): Primary 57R65; Secondary 57M99
- DOI: https://doi.org/10.1090/proc/13828
- MathSciNet review: 3754363