Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Gadgil, Siddhartha

Abstract (Summary of the Work)

This thesis deals with various questions regarding normal surfaces and Heegaard splittings of 3-manifolds.Chapter 1The first chapter is divided into two parts. In the first, we give an outline of normal surface theory and mention some of its important applications. The second part gives an overview of the theory of Heegaard splitting surfaces and a few of its applications. None of the material covered in this chapter is original and it is meant solely as an exposition of known results.Chapter 2In this chapter, we give a lower bound on the Euler characteristic of a normal surface, a topological invariant, in terms of the number of normal quadrilaterals in its embedding, obtained from its combinatorial description. A closed connected normal surface having no normal quadrilaterals is a vertex-linking sphere. We make the observation that a ‘strongly-connected’ normal surface (with boundary) having no normal quadrilaterals, is a planar surface. Using this fact, we obtain the desired relation. In the smooth category, we expect normal triangles to correspond to positive curvature pieces. Hence by Gauss-Bonnet, such a relation is to be expected when the curvature of the quadrilateral pieces is bounded below.Another result in a similar spirit is an upper bound on the number of normal triangles in terms of the number of normal quadrilaterals of a normal surface (having no vertex-linking spheres). Strongly-connected triangle components are shown to be subsets of vertex-linking spheres, so that the number of triangles in one such component is bounded above by the maximum number of triangles in a vertex-linking sphere. We think of quadrilaterals as ‘bridges’ linking the various strongly-connected triangle components and by a combinatorial argument we obtain the desired relation. Both these results pertain to original work published in the paper [43].Chapter 3Here we interpret a normal surface in a (singular) three-manifold in terms of the homology of a chain complex. This allows us to study the relation between normal surfaces and their quadrilateral co-ordinates. Specifically, we give a proof of an (unpublished) result of CassonRubinstein saying that quadrilaterals determine a normal surface up to vertex linking spheres. We also characterise the quadrilateral coordinates that correspond to a normal surface in a (possibly ideal) triangulation. The results in this chapter are the outcome of joint work with my adviser, Siddhartha Gadgil. They have been submitted as paper [45]Chapter 4We describe a procedure for refining the given triangulation of a 3-manifold that scales the PL-metric according to a given weight function while creating no new normal surfaces.It is known that an incompressible surface F in a triangulated irreducible 3-manifold M is isotopic to a normal surface that is of minimal PL-area in the isotopy class of F. Using the above scaling refinement we prove the converse. If F is a surface in a closed 3-manifold M such that for any triangulation τ of M, F is isotopic to a τ -normal surface F(τ ) that is of minimal PL-area in its isotopy class, then we show that F is incompressible. This is the result of original work and has been published as paper [44].


ProQuest Collection ID:

Control Number


Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.


Included in

Mathematics Commons