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One of the first ideas of topology is to study a complicated topological
space by decomposing it into simple blocks. The most important example
of a decomposition of this kind is triangulation, which is a
decomposition into simplices. If one has a triangulation of a
topological space, this makes him much easier to work with this space.
For example, to calculate the homology of this space, instead of
cumbersome groups of singular chains, one can take much more convenient
groups of simplicial chains. In addition, triangulation can be seen as a
way of encoding space for computer calculations. It turns out, however,
that there is the following surprising fact at first glance. Although it
was proved almost 90 years ago that any smooth manifold admits a
triangulation (Cairns, 1934), in dimensions 4 and higher, the list of
manifolds for which at least one explicit triangulation is known is
extremely small. Therefore, known explicit triangulations of relatively
simple manifolds are very interesting to study; as a rule, they are
beautiful combinatorial objects with interesting symmetry groups. Of
particular interest is Kühnel's remarkable 9-vertex triangulation of
the complex projective plane $\mathbb{CP}^2$ (1980), which essentially
started the science of explicit triangulations. Within the framework of
the lecture course, an introduction to the theory of triangulations of
manifolds will be given with an emphasis on the constructions of
explicit triangulations.
 RSS: Forthcoming seminars
 Lecturer
Gaifullin Alexander Aleksandrovich
Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center |
