Murat Saglam (Leipzig): A search for finer topological information via holomorphic curves in the symplectizations: the case of lens spaces and their unit cotangent bundle
Wednesday, 16.11.2016 16:00 im Raum N2
Abstract: It is classically known that two lens spaces L(p,q) and L(p,q')
and their products with S^2 are diffeomorphic if and only if q'\equiv \pm
q^{-\pm 1} in mod p. The non-trivial part of this statement is established
via the Reidemeister torsion.
In this talk, we explore the possibility of capturing the above relation
using holomorphic curves in the symplectizations and symplectic
cobordisms. We endow
lens spaces with the contact form that is the quotient of the standard
contact form on S^3 and in the case of unit cotangent bunndles, the
contact form is the quotient of the contact form on the unit cotangent
bundle of S^3 induced by the round metric. In both cases, the moduli
spaces of curves with at most two non-contractible ends are easy to be
described. Using these moduli spaces in a neck-stretching procedure, we
aim to show that given a contactomorphism between two lens spaces or two
unit cotangent bundles, the above relation is fulfilled. It turns out
that, this is not possible using the standard
methods, which is expected because of the theoretical background of the
problem.
After pointing out what goes wrong along the procedure, we show that
imposing conditions on the contactatomorphism solves both the technical
and essential issues of the procedure. In the case of unit cotangent
bundles, the conditions we impose are global bounds on the
contactomorphism, while in the case of lens spaces, the condition is the
strictness along a single contractible orbit.
Angelegt am 26.10.2016 von N. N
Geändert am 15.11.2016 von N. N
[Edit | Vorlage]