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RESEARCH STATEMENT My Area of Research Is Low-Dimensional RESEARCH STATEMENT My area of research is low-dimensional topology. In particular I am interested in triangulations/normal surfaces, Heegaard splittings, laminations and more recently contact structures/open book decomposions of 3-manifolds. 1. Taut Foliations A foliation of a 3-manifold is called taut if there exists an embedded simple closed curve in the manifold that intersects each leaf of the foliation transversally. The existence of taut-foliations in closed orientable 3-manifolds imply useful topological properties for the manifold. For example, if M is a closed, orientable 3-manifold that has a taut foliation with no sphere leaves then M is covered by R3, M is irreducible and has an infinite fundamental group. We have extended Roberts' theorem[7] for compact fibered 3-manifolds with one boundary component to the multiple boundary component case. Theorem 1.1 (Kalelkar-Roberts [11]). Given an orientable, fibered compact 3- manifold, a fibration with fiber surface of positive genus can be perturbed to yield taut foliations realizing a neighbourhood of boundary multi-slopes about the boundary multi-slope of the fibration. As a corollary, we obtain the following for for closed manifolds: Corollary 1.2 (Kalelkar-Roberts [11]). Let Mc(rj) be the closed manifold obtained j k from M by Dehn filling M along the multi-curve with rational multi-slope (r )j=1. When (rj) is sufficiently close to the multi-slope of the fibration, Mc(rj) also has a taut foliation. A contact structure is a 1-form on the 3-manifold satisfying a certain 'nowhere parallel' condition. There is a close relation between contact structures, open book decompositions and foliations, in particular both taut foliations and certain open- books perturb to tight contact structures. The closed manifold constructed in the corollary above has both a natural open book decomposition and taut foliation. We have shown that the contact structures obtained from both these constructs are isotopic. This result has been accepted for publication in the Pacific Journal of Math. 2. Heegaard splitting of Haken Manifolds A Heegaard splitting is a splitting of a 3-manifold into simpler pieces called han- dlebodies. Every closed 3-manifold has a Heegaard splitting, however this splitting may not be unique. There are manifolds with infinitely many non-isotopic strongly irreducible splittings. Tao Li [1] recently proved that non-Haken 3-manifolds have only finitely many irreducible Heegaard splittings up to isotopy [2]. Moriah, Schleimer and Sedgwick [3] have shown that for all known examples of manifolds with infinitely many irreducible splittings, the splittings are given by spinning (Haken summing) a fixed splitting surface about an incompressible surface. These results lead us to state the following conjecture. 1 2 RESEARCH STATEMENT Conjecture 2.1. Let M be a closed, orientable, irreducible and atoroidal 3-manifold, with infinitely many strongly irreducible Heegaard splittings. Then, there exists a strongly irreducible Heegaard splitting surface S and a sequence of (possibly discon- nected) incompressible surfaces Kn such that Sn = S + Kn is a sequence of strongly irreducible Heegaard splittings. In particular, instead of looking at limits in the space of projective measured laminations carried by a branched surface, we put a topology on the space of all laminations which makes it compact and where limits lie somewhere in between the limits produced by the Hausdorff-Gromov metric on the neighbourhood of the branched surface and the projective measured lamination limit. We then attempt to extend the methods employed by Tao Li in [2], for Haken manifolds. This project is currently impeded by a hard technical problem with laminations carried by branched surfaces, but considering the payoff, I would like to return to this when I have more time. 3. Triangulations 3.1. Euler Characteristic and quadrilaterals of a normal surface. We have given a lower bound on the Euler characteristic of a normal surface, a topologi- cal invariant, in terms of the number of normal quadrilaterals in its embedding, obtained from its combinatorial description. In the smooth category, we expect normal triangles to correspond to positive curvature pieces so such a result would be expected from Gauss-Bonnet theorem if the 'curvature' of quadrilateral pieces is bounded below. The exact theorem we have is the following: Theorem 3.1 (Kalelkar [8]). Let F be a normal surface in a triangulated 3- manifold M. Let Q be the number of normal quadrilaterals in F . Then, χ(F ) ≥ 2 − 7Q In particular, if F is an oriented, closed and connected normal surface of genus g, 7 g ≤ Q 2 3.2. A chain complex and quadrilaterals for normal surfaces. We have interpreted a normal surface in a (singular) three-manifold in terms of the homology of a chain complex. This allowed us to study the relation between normal surfaces and their quadrilateral co-ordinates. Specifically, we give another proof of a result of Casson-Rubinstein-Tollefson which says 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. Theorem 3.2 (Gadgil-Kalelkar [10], and Casson-Rubinstein-Tollefson[4]). Let ζ be an admissible, non-negative set of quadrilateral coordinates that can be represented by a normal surface. Then there is a set of admissible, non-negative normal surface coordinates ξ corresponding to ζ such that if ξ0 is another set of such coordinates, 0 P then ξ = ξ + v2V mv[S(v)], with mv ≥ 0 and S(v) the vertex linking sphere at v. RESEARCH STATEMENT 3 3.3. Incompressibility and normal minimal surfaces. A third result we have obtained is 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. Theorem 3.3 (Kalelkar [9]). Let F be a closed surface embedded in a 3-manifold M no component of which is a 2-sphere. Let τ be a triangulation of M. Let 0 f : f∆ : ∆ 2 τg ! Z be a scaling function and let τ = φf (τ) be the corresponding refinement of τ. Then, F is τ-normal , F is τ 0-normal. It is a well-known result of Haken that an incompressible surface F in a trian- gulated 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. Theorem 3.4 (Kalelkar [9]). Let F be a closed orientable surface in an irreducible orientable closed 3-manifold M. Then, F is incompressible if and only if for any triangulation τ of M, there exists a τ-normal surface F (τ) isotopic to F that is of minimal PL-area in the isotopy class of F . Rubinstein, Feng Lou and Stephen Tillmann have a program to obtain minimal and geometric triangulations, including recognition problems for a wide class of manifolds. The interplay between hyperbolic metrics and triangulations has not yet been well-studied and would be something I want to focus on. References 1. Tao Li, Heegaard surfaces and measured laminations, I: The Waldhausen Conjecture, Invent. Math. 167 (2007), no. 1, 135{177. 57N10 (57R30) 2. Tao Li, Heegaard surfaces and measured laminations, II: Non-Haken 3-maniolds, J. Amer. Math. Soc. 19 (2006), no. 3, 625{657 (electronic).57N10 (57M25 57M50) 3. Yoav Moriah, Saul Schleimer, Eric Sedgwick, Heegaard splittings of the form H +nK, Comm. Anal. Geom. 14 (2006), no. 2, 215{247. 57N10 (57M25) 4. Jeffrey Tollefson, Normal surface Q-theory, Pacific J. Math. 183 (1998), no. 2, 359{374. 57N10 (57Q35) 5. Nathan Dunfield, Anil Hirani, The least spanning area of a knot and the optimal bounding chain problem., Computational geometry (SCG'11), 135144, ACM, New York, 2011. 6. Ian Agol, Joel Hass, William Thurston, 3-manifold knot genus is NP-complete., Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, 761766, ACM, New York, 2002. 7. Rachel Roberts, Taut foliations in punctured surface bundles. I., Proc. London Math. Soc. (3) 82 (2001), no. 3, 747768. 8. Tejas Kalelkar, Euler characteristic and quadrilaterals of normal surfaces; Proceedings Math- ematical Sciences, Indian Academy of Science, Volume 118, Number 2 / May, 2008; DOI: 10.1007/s12044-008-0015-7 9. Tejas Kalelkar, Incompressibility and normal minimal surfaces; Geometriae Dedicata, Volume 142, Number 1 / October, 2009, 61{70; DOI: 10.1007/s10711-009-9358-1 10. Siddhartha Gadgil and Tejas Kalelkar, A Chain complex and Quadrilaterals for normal sur- faces; Rocky Mountain Journal of Mathematics, Volume 43, Number 2, 2013 11. Tejas Kalelkar and Rachel Roberts, Taut foliations in surface bundles with multiple punctures, arXiv:1211.3637, Accepted for publication in the Pacific Journal of Math.
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