Local invariants of fronts in 3-manifolds



Alsaeed, Suliman
(2014) Local invariants of fronts in 3-manifolds. PhD thesis, University of Liverpool.

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Abstract

An invariant is a quantity which remains unchanged under certain classes of transformations. A wave front (or a front) in a 3-manifold is the image of a surface under a Legendrian map. The aim of this thesis is the description of all local invariants of fronts in 3-manifolds. The front invariants under consideration are those whose increments in generic homotopies are determined entirely by diffeomorphism types of local bifurcations of the fronts. Such invariants are dual to trivial codimension 1 cycles supported on the discriminant in the space of corresponding Legendrian maps. We describe the spaces of the discriminantal cycles (possibly non-trivial) for various orientation and co-orientation settings of the fronts in an arbitrary oriented 3-manifold, both for the integer and mod2 coefficients. For the majority of these cycles we find homotopy-independent interpretations which guarantee the triviality required. In particular, in the case of framed fronts we show that all integer local invariants of Legendrian maps without corank 2 points are essentially exhausted by the numbers of points of isolated singularity types of the fronts.

Item Type: Thesis (PhD)
Additional Information: Date: 2014-11 (completed)
Subjects: Q Science > QA Mathematics
Depositing User: Symplectic Admin
Date Deposited: 22 Jan 2016 11:30
Last Modified: 09 Jan 2021 08:50
DOI: 10.17638/02006756
Supervisors:
  • Goryunov, Victor
  • Pratoussevitch, Anna
URI: https://livrepository.liverpool.ac.uk/id/eprint/2006756