Homology Curve Complex

Ningthoujam Jiban Singh, Himadri Kumar Mukerjee

Department of Mathematics, Assam University, Silchar, India.
Department of Mathematics, North-Eastern Hill University, NEHU Campus, Shillong, India.

DOI: 10.4236/apm.2012.22017                

Abstract

A homological analogue of curve complex of a closed connected orientable surface is developed and studied. The dis-tance in this complex is shown to be quite computable and an algorithm given (Theorem 3.5). As an application of this complex it is shown that for a closed orientable 3-manifold, and any of its Heegaard splittings, one can give an algorithm to decide whether the manifold contains a 2-sided, non-separating, closed incompressible surface (Theorem 1.1).

Keywords

Homology Curve Complex; Heegaard Splittings; Incompressible Surface

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	W. Jaco and U. Oertel, “An Algorithm to Decide If a 3- Manifold Is a Haken Manifold,” Topology, Vol. 23, No. 2, 1984, pp. 195-209. doi:10.1016/0040-9383(84)90039-9
[2]	W. Haken, “Theorie der Normalflachen,” Acta Mathematica, Vol. 105, No. 3-4, 1961, pp. 245-375. doi:10.1007/BF02559591
[3]	W. Jaco, D. Letscher and J. H. Rubinstein, “Algorithms for Essential Surfaces in 3-Manifolds,” Contemporary Mathematics, Vol. 314, 2002, pp. 107-124. doi:10.1090/conm/314/05426
[4]	J. Birman, “The Topology of 3-Manifolds, Heegaard Distances and the Mapping Class Group of a 2-Manifold,” arXiv.org, 2005. http://arxiv.org/abs/math/0502545
[5]	I. Irmer, “Geometry of the Homology Curve Complex,” arXiv.org, 2011. http://arxiv.org/abs/1107.3547
[6]	J. Johnson and T. Patel, “Generalized Handlebody Sets and Non-Haken 3-Manifolds,” Pacific Journal of Mathematics, Vol. 235, No. 1. 2005, pp. 35-41.
[7]	W. Jaco, “Lectures on Three-Manifold Topology,” CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, 1980.
[8]	J. Hempel, “3-Manifolds as Viewed from the Curve Complex,” Topology, Vol. 40, No. 3,m 2001, pp. 631-657. doi:10.1016/S0040-9383(00)00033-1
[9]	D. Johnson, “An Abelian Quotient of the Mapping Class Group Ig,” Mathematische Annalen, Vol. 249, No. 3, 1980, pp. 225-242. doi:10.1007/BF01363897
[10]	M. D. Meyerson, “Representing Homology Classes of Closed Orientable Surfaces,” Proceedings of AMS, Vol. 61, No. 1, 1976, pp. 181-182.
[11]	M. Schaefer, E. Sedgwick and D. Stefankovic, “Algorithms for Normal Curves and Surfaces,” Lecture Notes in Computer Science, Springer, 2002, New York, pp. 370- 380.
[12]	F. C. Lei, “Complete Systems of Surfaces in 3-Mani-folds,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 122, No. 1, 1997, pp. 185-191. doi:10.1017/S0305004196001545
[13]	D. Saunders and Z. D. Wan, “Smith Normal Form of Dense Integer Matrices, Fast Algorithms into Practice,” Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation, 4-7 July 2004, Santander.