COMB3DTOP · Use of combinatorial methods in various approaches to three-dimensional topology

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Project objective

Since the work of Poi care, topologists have been looking for a comprehensive classification of three-dimensional manifolds and links. Recently, this quest has enjoyed spectacular developments based on various approaches to three-dimensional topology, namely contact topology, hyperbolic geometry and quantum topology with its cousin, the theory of finite-type invariants. Only little is known about the relationship between these independent theories. Nevertheless, according to work of Bernadette and Petrolia, each of these theories can be a tackled using standard spine of three-dimensional manifolds. Standard spines provide combinatorial methods to present three-dimensional manifolds as well as the additional structures with which they can be equipped. This methodology has already proved to be fruitful. Therefore, one could expect the theory of standard spines to become the node of some connections between hyperbolic geometry; contact topology and quantum topology in dimension three. The proposed project looks towards this perspective and, in the shorter term, addresses two problems pointing in that direction. The first of these problems consists in investigating connections between the theory of standard spines and the recent theory of finite-type invariants of compact oriented three-dimensional manifolds, which is due to Goussarov and Habra. A similar problem stands for refinements of these theories to manifolds equipped with Spin-structures or Eller structures. The second subject is that of Reidemeister-Turaev torsion of three-dimensional manifolds endowed with Eller structures. One interesting question is to compute the multiplicative variation of the torsion invariant under appropriate surgery modifications of the manifold. These computations, which have been started in Massuyeau? S thesis could be completed thanks to some extensions of this invariant due to Bernadette and Petrolia. Again, their techniques involve the use of standard spines.

Original text from CORDIS.

Participants

• UNIVERSITA DEGLI STUDI DI PISA · PISACoordinatorItaly

Links

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