Algebraic Topology

Winter 2019, Grade: 1.0

Despite its name, algebraic topology (AT) is more than topology with algebraic methods. While point-set topology is concerned with very fine properties (separability, completeness, countability, convergence, ...) of spaces, AT deals with more global phenomena such as (co-)homology, top. K-theory, homotopy, and characteristic classes. Locally, the spaces considered usually look very similar, such as for (smooth) manifolds, vector bundles of such, CW complexes, or normed algebras, but their global topology does not coincide.

History: The origins of the field can be traced back to Euler and his considerations of polyhedra. The Euler characteristic (1758) can be seen as the first topological invariant of a space. Major developments happened in the first half of the 20th century when H. Poincare constructed what is today known as simplicial homology for polyhedra. In the subsequent years, this notion was extended to more general spaces, ultimately culminating in singular homology theory introduced by S. Eilenberg. Together with N.E. Steenrod, he eventually proposed an axiomatised theory of (co-) homology, and together with S. Mac Lane, he laid the foundation for category theory in order to study various (co-)homology theories.

Inner workings: The inner workings of the theory mostly revolve around identifying topological or homotopical invariants that can be used to distinguish space and analysing these with algebraic methods (such as homological and homotopical algebra). Since being an invariant essentially means being a functor, the theory lends itself very well to category theoretical methods. However, in many classical aspects AT uses combinatorial methods (such as simplicial sets and complexes) to simplify classification problems (e.g. homotopy can be studied entirely on CW complexes).

Application in Pure Mathematics: AT finds application where ever topological spaces need to be distinguished by their global properties. Since topological spaces and classification problems are so ubiquitous in mathematics, this happens quite often. Concrete examples are algebraic (e.g. sheave cohomology, l-adic cohomology) and differential geometry (e.g. Chern classes,  deRham cohomology) or operator algebras (e.g. operator K-theory).

Application in Applied Mathematics: Today, mathematical physics makes great use of AT in that many physical problems, especially in fundamental physics can be realized topologically. Classical results on de Rham cohomology are used electromagnetism, whereas modern methods in AT are used in quantum field theories. More recent applications include topological data analysis, where (persistent) homology can be used to identify clusters in data or to recover shapes from incomplete information.