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Çagri Karakurt (Bogaziçi University, Istanbul): Surgery obstructions and Heegaard Floer homology

ABSTRACT: A classical theorem of Lickorish and Wallace says that every
three-manifold can be obtained by a Dehn surgery on a link, but this
surgery representation is not unique. In fact the whole machinery of
Kirby calculus would give infinitely many different surgery pictures of
the same three-manifold. A natural question here is whether one can
always use the tools of Kirby calculus to make the number of link
components in a surgery representation of a three-manifold to be one.
This question is especially subtle if the three-manifold is an
irreducible homology sphere. In fact, until recently the only known
obstruction to being surgery on a knot was found 20 years ago by Auckly
who used a hard gauge theoretical result of Taubes. Recently in a joint
work with J. Hom and T. Lidman, we found a more elementary obstruction
which uses Heegaard Floer homology. In this talk, I.ll mention the
essentials of this work and give infinitely many examples of Brieskorn
spheres which are not surgery on a knot.