Low Dimensional Topology


The project started in June 2010 (as the "Lendulet" research group Low Dimensional Topology of HAS) and was enhanced by the ERC Grant LDTBud in April 2012.

Our main fields of study

Primary focus of interests

Heegaard Floer theory. In this project we plan to extend our earlier results computing various versions of Heegaard Floer homologies purely combinatorially. We also plan to find combinatorial definitions of these invariants (as graded groups). Such results will potentially lead to a combinatorial description of 4-dimensional Heegaard Floer (mixed) invariants, conjecturally equivalent to Seiberg-Witten invariants of smooth 4-manifolds. In particular, we hope to find a combinatorial proof of Donaldsonâ..s diagonalizability theorem, and find relations between the Heegaard Floer and the fundamental groups of a 3-manifold.

Contact topology. Using Heegaard Floer theory and contact surgery, a systematic study of existence of tight contact structures on 3-manifolds (in particular, on hyperbolic 3-manifolds) is planned. The verification of the existence of tight structures on 3-manifolds given by surgery on a knot (with high enough framing) in the 3-sphere is proposed. Similar techniques also apply in studying Legendrian and transverse knots in contact 3-manifolds. Using the Legendrian invariant of knots, Legendrian and transverse simplicity can be conveniently studied. Parallel to these efforts, we examine the existence question of contact structures on high dimensional manifolds. Using surgery theory and Eliashberg's celebrated results concerning Stein manifolds, we plan to develope methods for finding Stein fillable contact structures and identify obstructions for the existence of such.

Exotic 4-manifolds. Extending our previous results, we plan to investigate the existence of exotic smooth structures on 4-manifolds with small Euler characteristics, such as the complex projective plane CP2, its blow-up CP2#CP2-bar, the product of two complex projective lines CP1Ă.CP1 and ultimately the 4-dimensional sphere S^4. Possible extensions of the rational blow down procedure (successful in producing exotic structures) will be also studied. These ideas lead to the study of rational homology disk smoothings of normal surface singularities and applications of symplectic methods in the theory of surface singularities.


The group has members in the doctoral, post-doctoral and senior researcher level.

Principal Investigator:

Doctoral students:

Postdoctoral fellows:

Senior Reserachers:

Short time visitors:


Conferences held:

Related Publications:

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