DOI QR코드

DOI QR Code

HOMOLOGY 3-SPHERES OBTAINED BY SURGERY ON EVEN NET DIAGRAMS

Lee, Sang-Youl

  • Received : 2011.10.28
  • Published : 2012.10.31

Abstract

In this paper, we characterize surgery presentations for $\mathbb{Z}$-homology 3-spheres and $\mathbb{Z}/2\mathbb{Z}$-homology 3-spheres obtained from $S^3$ by Dehn surgery along a knot or link which admits an even net diagram and show that the Casson invariant for $\mathbb{Z}$-homology spheres and the ${\mu}$-invariant for $\mathbb{Z}/2\mathbb{Z}$-homology spheres can be directly read from the net diagram. We also construct oriented 4-manifolds bounding such homology spheres and find their some properties.

Keywords

Casson invariant;Dehn surgery;homology 3-sphere;${\mu}$-invariants;net diagram;4-manifold

References

  1. S. Akbulut and J. McCathy, Casson's Invariant for Oriented Homology 3-Spheres, Princeton Math. Notes 36, Princeton University Press, 1990.
  2. J. Eells and N. Kuiper, An invariant for certain smooth manifolds, Ann. Mat. Pura Appl. (4) 60 (1962), 93-110. https://doi.org/10.1007/BF02412768
  3. R. A. Fenn and C. Rourke, On Kirby's calculus of links, Topology 18 (1979), no. 1, 1-15. https://doi.org/10.1016/0040-9383(79)90010-7
  4. H. M. Hilden, Representations of homology 3-spheres, Pacific J. Math. 94 (1981), no. 1, 125-129. https://doi.org/10.2140/pjm.1981.94.125
  5. F. Hirzebruch, W. Neumann, and S. Koh, Differentiable Manifolds and Quadratic Forms, Marcel Dekker, 1971.
  6. R. Kirby, A calculus for framed links in $S^{3}$, Invent. Math. 45 (1978), no. 1, 35-56. https://doi.org/10.1007/BF01406222
  7. S. Y. Lee and M. Seo, Formulas for the Casson invariant of certain integral homology 3-spheres, J. Knot Theory Ramifications 18 (2009), no. 11, 1551-1576. https://doi.org/10.1142/S0218216509007610
  8. W. B. R. Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. Math. 76 (1962), 531-540. https://doi.org/10.2307/1970373
  9. S. Matveev, Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki 42 (1987), no. 2, 268-278, 345.
  10. V. Rokhlin, New results in the theory of four-dimensional manifolds, Doklady Acad. Nauk SSSR 84 (1952), 221-224.
  11. D. Rolfsen, Rational surgery calculus: extension of Kirby's theorem, Pacific J. Math. 110 (1984), no. 2, 377-386. https://doi.org/10.2140/pjm.1984.110.377
  12. D. Rolfsen, Knots and Links, Publish or Perish, 1990.
  13. N. Saveliev, Lectures on the Topology of 3-Manifolds, de Gruyter, 1999.
  14. N. Saveliev, Invariants for Homology 3-Spheres, Springer, 2002.
  15. A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503-528. https://doi.org/10.4153/CJM-1960-045-7

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)