86 research outputs found
Contact Dehn surgery, symplectic fillings, and Property P for knots
These are notes of a talk given at the Mathematische Arbeitstagung 2005 in
Bonn. Following ideas of Ozbagci-Stipsicz, a proof based on contact Dehn
surgery is given of Eliashberg's concave filling theorem for contact
3-manifolds. The role of that theorem in the Kronheimer-Mrowka proof of
property P for nontrivial knots is sketched.Comment: 9 page
-plumbings and exotic contact structures on spheres
We prove the existence of exotic but homotopically trivial contact structures
on spheres of dimension 8k-1. Together with previous results of Eliashberg and
the second author this establishes the existence of such structures on all
odd-dimensional spheres (of dimension at least 3).Comment: 12 page
Transversely holomorphic flows and contact circles on spherical 3-manifolds
Motivated by the moduli theory of taut contact circles on spherical
3-manifolds, we relate taut contact circles to transversely holomorphic flows.
We give an elementary survey of such 1-dimensional foliations from a
topological viewpoint. We describe a complex analogue of the classical
Godbillon-Vey invariant, the so-called Bott invariant, and a logarithmic
monodromy of closed leaves. The Bott invariant allows us to formulate a
generalised Gau{\ss}-Bonnet theorem. We compute these invariants for the
Poincar\'e foliations on the 3-sphere and derive rigidity statements, including
a uniformisation theorem for orbifolds. These results are then applied to the
classification of taut contact circles.Comment: 31 pages, 3 figures; v2: changes to the exposition, additional
reference
A formula for the Chern classes of symplectic blow-ups
It is shown that the formula for the Chern classes (in the Chow ring) of
blow-ups of algebraic varieties, due to Porteous and Lascu-Scott, also holds
(in the cohomology ring) for blow-ups of symplectic and complex manifolds. This
was used by the second-named author in her solution of the geography problem
for 8-dimensional symplectic manifolds. The proof equally applies to real
blow-ups of arbitrary manifolds and yields the corresponding blow-up formula
for the Stiefel-Whitney classes. In the course of the argument the topological
analogue of Grothendieck's `formule clef' in intersection theory is proved.Comment: 19 page
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