Mohan Bhupal

Symplectic fillings of links of quotient surface singularities

We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.

Canonical contact structures on some singularity links

We identify the canonical contact structure on the link of a simple elliptic or cusp singularity by drawing a Legendrian handlebody diagram of one of its Stein fillings. We also show that the canonical contact structure on the link of a numerically Gorenstein surface singularity is trivial considered as a real plane bundle.

Symplectic fillings of lens spaces as Lefschetz fibrations

We construct a positive allowable Lefschetz fibration over the disk on any minimal weak symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding complex two-dimensional cyclic quotient singularity.

Smoothings of Singularities and Symplectic Topology

We review the symplectic methods which have been applied in the classification of weighted homogeneous singularities with rational homology disk (√HD) smoothings. We also review the construction of such smoothings and show that in many cases these smoothings are unique up to symplectic deformation. In addition, we describe a method for finding differential topological descriptions (more precisely, Kirby diagrams) of the smoothings and illustrate this method by working out a family of examples.

OPEN BOOK DECOMPOSITIONS OF LINKS OF SIMPLE SURFACE SINGULARITIES

We describe open book decompositions of links of simple surface singularities that support the corresponding unique Milnor fillable contact structures. The open books we describe are isotopic to Milnor open books.

On symplectic fillings of links of rational surface singularities with reduced fundamental cycle

We prove that every symplectic filling of the link of a rational surface singularity with reduced fundamental cycle admits a rational compactification, possibly after a modification of the filling in a collar neighbourhood of the link.

A generalisation of the Morse inequalities

In this paper we construct a family of variational families for a Legendrian embedding, into the 1-jet bundle of a closed manifold, that can be obtained from the zero section through Legendrian embdeddings, by discretising the action functional. We compute the second variation of a generating funciton obtained as above at a nondegenerate critical point and prove a formula relating the signature of the second variation to the Maslov index as the mesh goes to zero. We use this to prove a generlisation of the Morse inequalities thus refining a theorem of Chekanov.