Roman Nedela
The homeomorphism problem on 3-manifolds attracts mathematicians since the
beginnings of 20-th century.
A particular but fundamental instance of it is known as the Poincare
conjecture. Its central position
in the core of classical mathematics is stressed by the fact that a
formulation of the problem belongs
to the list of the seven millennium problem proclaimed by the Clay Institute
of Mathematics.
The aim of the talk is to show how some classical mathematical problems can be
formulated in a combinatorial
way. One of the advantages of this approach is a possibility to use computer
to study `small' examples.
The most developed part of a combinatorial approach is knot theory based on a
well-known fact
that any 3-manifold can be constructed as a branched cover over the 3-sphere,
where the
set of branch points forms a knot. In our talk we present another, not so
known combinatorial approach to 3-manifolds. We first introduce a
combinatorial counterpart of the classical homeomorphism problem on 3-manifolds
using a theory built by Pezzana and his successors. Pezzana proved that every
closed compact orientable 3-manifold can be represented by a
bipartite 4-edge-coloured 4-valent graph called a crystallisation of
.
Crystallisation is a 4-valent 4-edge-coloured graph such that removal of any
monochromatic (perfect) matching yields a planar graph. This way
3-dimensional objects are replaced by 1-dimensional which allows us to employ
combinatorial
methods to study the homeomorphism problem.
We explain a result of Casali and Grasselli showing that 3-manifolds of
Heegaard
genus
can be represented by crystallisations with a very simple structure
which can be described by
a
-tuple of non-negative integers. The sum of first
integers is
called complexity of the
admissible
-tuple. If
is the complexity then the number of
vertices of the
associated graph is
. We give a combinatorial definition of the Heegaard
genus and show how
to classify 3-manifolds of Heegard at most one using graph theoretical
results. Then we introduce
an algorithm to derive a presentation of the fundamental group of a 3-manifold
from the representation
by means of a
-tuple of integers. A 3-manifold is called prime if it
cannot be expressed
as a connected sum of non-trivial 3-manifolds.
We show how to classify all prime 3-manifolds of Heegaard genus 2 described by
6-tuples of complexity at most 21.
Finally, we add some remarks on the celebrated Thurston's geometrisation
conjecture and on recent
progress in the topic.