186 research outputs found
Algorithmic aspects of branched coverings
This is the announcement, and the long summary, of a series of articles on
the algorithmic study of Thurston maps. We describe branched coverings of the
sphere in terms of group-theoretical objects called bisets, and develop a
theory of decompositions of bisets.
We introduce a canonical "Levy" decomposition of an arbitrary Thurston map
into homeomorphisms, metrically-expanding maps and maps doubly covered by torus
endomorphisms. The homeomorphisms decompose themselves into finite-order and
pseudo-Anosov maps, and the expanding maps decompose themselves into rational
maps.
As an outcome, we prove that it is decidable when two Thurston maps are
equivalent. We also show that the decompositions above are computable, both in
theory and in practice.Comment: 60-page announcement of 5-part text, to apper in Ann. Fac. Sci.
Toulouse. Minor typos corrected, and major rewrite of section 7.8, which was
studying a different map than claime
Topological fluid mechanics of point vortex motions
Topological techniques are used to study the motions of systems of point
vortices in the infinite plane, in singly-periodic arrays, and in
doubly-periodic lattices. The reduction of each system using its symmetries is
described in detail. Restricting to three vortices with zero net circulation,
each reduced system is described by a one degree of freedom Hamiltonian. The
phase portrait of this reduced system is subdivided into regimes using the
separatrix motions, and a braid representing the topology of all vortex motions
in each regime is computed. This braid also describes the isotopy class of the
advection homeomorphism induced by the vortex motion. The Thurston-Nielsen
theory is then used to analyse these isotopy classes, and in certain cases
strong conclusions about the dynamics of the advection can be made
Homology cylinders: an enlargement of the mapping class group
We consider a homological enlargement of the mapping class group, defined by
homology cylinders over a closed oriented surface (up to homology cobordism).
These are important model objects in the recent Goussarov-Habiro theory of
finite-type invariants of 3-manifolds.
We study the structure of this group from several directions: the relative
weight filtration of Dennis Johnson, the finite-type filtration of
Goussarov-Habiro, and the relation to string link concordance.
We also consider a new Lagrangian filtration of both the mapping class group
and the group of homology cylinders.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-12.abs.htm
Combinatorics of tight geodesics and stable lengths
We give an algorithm to compute the stable lengths of pseudo-Anosovs on the
curve graph, answering a question of Bowditch. We also give a procedure to
compute all invariant tight geodesic axes of pseudo-Anosovs.
Along the way we show that there are constants such that the
minimal upper bound on `slices' of tight geodesics is bounded below and above
by and , where is the complexity of the
surface. As a consequence, we give the first computable bounds on the
asymptotic dimension of curve graphs and mapping class groups.
Our techniques involve a generalization of Masur--Minsky's tight geodesics
and a new class of paths on which their tightening procedure works.Comment: 19 pages, 2 figure
Combinatorial cohomology of the space of long knots
The motivation of this work is to define cohomology classes in the space of
knots that are both easy to find and to evaluate, by reducing the problem to
simple linear algebra. We achieve this goal by defining a combinatorial graded
cochain complex, such that the elements of an explicit submodule in the
cohomology define algebraic intersections with some "geometrically simple"
strata in the space of knots. Such strata are endowed with explicit
co-orientations, that are canonical in some sense. The combinatorial tools
involved are natural generalisations (degeneracies) of usual methods using
arrow diagrams.Comment: 20p. 9 fig
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