Geometric
topology is often split into low dimensional (4 or less)
and high dimensional. This split is based upon the techniques
employed, the kinds of question that can be answered,
and the state of knowledge. There were enormous advances
in high dimensional topology during the 60’s including
the solution of the high dimensional Poincare conjecture,
and a good understanding of how differentiability enters
into the picture, for example through the existence of
exotic smooth structures on spheres.
Today
a considerable effort is being made to better understand
manifolds of dimensions 3 and 4. The techniques, conjectures
and outlooks in these two areas are very different, although
there have also been hints of various unifying themes.
In the 80’s it was discovered by Donaldson, Freedman
and Casson that Euclidean space has exotic smooth structures
only in 4 dimensions.
The
theory of 3 dimensional manifolds was revolutionized in
the late 70’s by Thurston’s Geometrization Conjecture.
There are eight geometries (homogeneous Riemannian metrics)
which (appear to) play a similar role in 3 dimensions
to the three constant curvature geometries in two dimensions.
Some problems in 3dimensions are best studied through
combinatorial and topological techniques using surfaces
and their generalizations. Many problems in knot theory
are of this type.
There
are many connections to number theory, Riemannian geometry,
geometric group theory and dynamical systems to name only
a few.
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