Low-dimensional Topology

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Description

Tuesday 09.00-11.00

This course will be a general overview of the geometry and topology of low-dimensional manifolds, i.e., manifolds of dimensions 2 and 3. Our plan is as follows:

We will first review constructions and classification of surfaces (2-manifolds) and discuss their mapping class group, i.e., their “topological symmetry group”.

We will then turn our attention to geometry and construct hyperbolic metrics on surfaces, briefly touching upon Teichmüller space and moduli space, which are parameter spaces for these metrics.

After this we jump up a dimension to 3-manifolds and show how algebraic structures of the mapping class group encode information about surfaces and 3-manifolds.

We will also discuss various constructions of 3-manifolds and describe canonical ways to split them into so-called geometric pieces, ending with the statement of Thurston’s Geometrisation Conjecture.