Geometric Topology

Siddhartha GadgilDepartment of Mathematics,Indian Institute of ScienceGeometric Topology: Low-dimensional manifolds, groups and minimal surfaces

1Geometric TopologyTopology is the study of global properties.Formally, we study properties invariant under homeomorphism, diffeomorphism etc.A powerful approach is Algebraic Topology, where we associate algebraic problems to topological ones.Geometric Topology concerns the relation between topology and the associated algebra (or analysis).Sometimes geometric methods are used to study algebraic problems using this correspondence.

2Intersections of Curves on SurfacesProblem: Given curves on a surface, deform them continuously (by a homotopy) so thatAll points of intersection are transversal.The number of intersection points is minimised.

3Intersection of Curves: Algebraic topologyWe can count the intersection points with sign.This is invariant under deformation.Hence we get a lower bound on the number of intersection points.

4Geometric topology: Whitney discsIf there is an embedded disc with one arc in each curve, then we can cancel intersection points.The existence of a disc that is allowed to cross itself is determined by algebraic topology.

5High-dimensional topologyIn dimensions five and above, (two-dimensional) discs are generically embedded.This allows topology to be reduced to algebra using surgery, handle-cancellation etc.However, the corresponding algebraic questions may be very difficult.In dimensions three and four, generic discs are not embedded due to knotting.

6Dimensions 2 and 3In dimensions 2 and 3, topology is related to geometric structures.The simplest instance of this is the uniformisation theorem for surfaces.The climax of such a viewpoint is Perelmans proof of Thurstons geometrization conjecture.The algebraic topology is determined in these cases by the fundamental group.Hence we get relations between topology, geometry and group theory.

7Dimension 4Due to the work of Freedman, topological four-manifolds up to homeomorphism essentially behave like high-dimensional manifolds.However, since the work of Donaldson, it is known that smooth four-manifolds behave very differently.Major advances have been made due to the invariants of Donaldson, Seiberg-Witten and Ozsvath-Szabo.However, the differential topology of four-manifolds remains mysterious.

8Homology and intersection pairingFor a non-orientable surface, the intersection pairing gives a pairing on homology with value in integers modulo 2.Any homeomorphism of surfaces preserves this.We showed that the converse is true.

S. Gadgil and D. Pancholi, Homology and homeomorphisms of non-orientable surfaces, Proc. Indian Acad. Sci. Math. Sci. vol. 115 p. 251 (2005).

9Involutions and theta characteristics

I. Biswas, S. Gadgil, and P. Sankaran, On theta characteristics of a compact Riemann surface Bull. Sci. Math. 131,no. 5, p. 493 (2007).I. Biswas, S. Gadgil, Real theta characteristics and automorphisms of a real curve, submitted for publication

10Hidden Elliptic operatorsOften, elliptic operators not a priori associated to a problem have major consequences.It is thus important to be able to find elliptic operators that may not be evident.

S. Gadgil, Limits of functions and elliptic operators Proc. Indian Acad. Sci. Math. Sci. vol. 114, p. 153 (2004).

11Geodesics minimise intersectionThe problem of minimisation of intersection can be fruitfully approached using geometric ideas.Namely, we consider geodesics, i.e., curves that are critical points (for instance minima) for length.On a hyperbolic surface, such geodesics automatically intersect minimally.For a compact (hyperbolic) surface, every curve can be deformed to a (unique) geodesic.By analogy with geodesics, Scott and Swarup have defined algebraic intersection numbers for groups.

12Least area surfacesLeast area surfaces are surfaces that minimise area in their homotopy class.By the Meeks-Yau trick, such surfaces in 3-manifolds intersect minimally.However, the existence, of least area surfaces is a subtle question due to the phenomenon of bubbling.Fundamental results of Sacks-Uhlenbeck and Schoen-Yau assert that, for any Riemannian metric, any incompressible surface (in an irreducible manifold) is homotopic to a least area surface.

13Incompressibility and least area surfacesWe showed that incompressibility is characterised by the existence of least-area surfaces.Let F be a surface in a smooth three-manifold M.

S. Gadgil, Incompressibility and Least-area surfaces, Expositiones Mathematicae 26, p.93 (2008).

14Compactness for bounded mean curvatureMinimal surfaces (including least area surfaces) are characterised by having mean curvature zero.We study limits of surfaces of bounded mean curvature, with an upper bound on area and genus.We prove a compactness theorem in a C0 sense.We show that the limit is two-dimensional in an appropriate sense.S. Gadgil and H. Seshadri, A compactness theorem for surfaces of bounded mean curvature in Riemannian manifolds, preprint.

15Representing by embedded spheresFree groups can be studied in terms of the manifold

We gave an algorithm to decide when a sphere in this manifold can be deformed to an embedded sphere.We also gave an algorithm to decide when such a collection of spheres can be deformed to be disjoint.The analogous question for any orientable manifold can be reduced to the case of M.

S. Gadgil, Embedded spheres in S2 S1 #...# S2 S1, Topology Appl. vol. 153, p. 1141 (2006).

16Algebraic & Geometric intersection numbersFor spheres in the manifold we have two notions of intersection numbers:

We also obtain the following:

S. Gadgil and S. Pandit, Algebraic and Geometric intersection numbers for free groups, submitted for publication.

17Ordering 3-manifoldsClosed, oriented surfaces are determined by their genus.

Hence there is a natural order on surfaces.This order can be generalised to higher dimensions to one based on the existence of degree one maps.This is however a partial order in higher dimensions.This order is particularly fruitful for 3-manifolds.

18Surgery on 3-manifoldsDehn surgery is the process of deleting a regular neighbourhood of a knot (or link) in a 3-manifold and gluing in solid tori in a different way.

The Lickorish-Wallace theorem says that for any pair of closed oriented manifolds M and N, M can be obtained from N by surgery about some link in N.

19Degree-one maps and surgeryWe characterise the existence of degree-one maps in terms of surgery.

S. Gadgil, Degree-one maps, Surgery and four-manifolds, Bull. Lond. Math. Soc. 39, p. 419 (2007)

20Exotic smooth structuresExotic manifolds are manifolds that are homeomorphic but not diffeomorphic to standard spaces.A revolutionary result of Minor was the construction of exotic spheres in dimension 7.Exotic structures in high dimensions can be understood completely in algebraic terms.However, the work of Freedman and Donaldson resulted in the construction of exotic R4s, which show behaviour very different from high dimensions.

21Field theories and Exotic structures on R4The exotic R4s were shown to be not diffeomorphic to R4 by indirect arguments using closed manifolds.We constructed an invariant of certain open manifolds using field theoretic properties of Ozsvath-Szabo invariants.We showed that our invariants are strong enough to detect exotic R4s.S. Gadgil, Open manifolds, Ozsvath-Szabo invariants and Exotic R4s, submitted for publication.

22One-handles for 4-manifoldsA central technique in high-dimensional topology is handle-cancellation, due to Smale.In particular, this asserts that simply-connected manifolds have handle-decompositions without one-handles.We showed that this is not true in dimension 4 in a strong relative sense, in a manner sensitive to the smooth structure.

S. Gadgil, One-handles and Concordance Kernels for smooth 4-manifolds, submitted for publication

23Extremes of the Indian MonsoonEl Nino (ENSO) has been long known to influence the Indian Summer Monsoon Rainfall.In recent years, another factor (EQWIN) related to the Indian Ocean has been found to be significant. Using order statistics, we showed that indices for EQWIN and ENSO together are very strongly related to extremes of the monsoon.There is no relation to rainfall in the normal range.Sulochana Gadgil, P.N. Vinaychandran and P.A. Francis and Siddhartha Gadgil, Extremes of the Indian summer monsoon, Geophysical Research letters, vol. 31, no. 12, (2004).

24Watson-Crick pairing of RNA Ribose Nucleic Acid (RNA) plays a very important role in living systems.The properties of an RNA molecule depend on the way it folds. In particular the Watson-Crick pairing of bases plays a central role.Watson-Crick pairing can be naturally modelled in terms of free groups.

25Milnor invariants and Allosteric structuresWe introduced methods from topology, namely Milnor invariants of links, to understand the folding of RNA molecules due to Watson-Crick pairing.We showed that the first of the new invariants introduced constructed can predict allosteric structures, i.e., deep local minima.Allosteric structures are of fundamental importance in enzyme regulation.

S. Gadgil, Watson-Crick pairing, the Heisenberg group and Milnor invariants, submitted for publication.

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