Module 24-M-GT-AT1 Algebraic Topology 1

Non-official translation of the module descriptions. Only the German version is legally binding.

Students master the basic contents and methods of Algebraic Topology, in particular they can independently carry out complex proofs in this area requiring a high level of mathematical expertise. They acquire the techniques for doing substantial computations with homology, learn about fundamental properties of Euclidean space and acquire basic familiarity with simplicial sets. Concretely:

• Students are able to define central concepts of the theory (e.g. singular simplicies, cell complexes, fundamental classes) and use these in context
• Students know applications of the theory (e.g. Jordan’s separation theorem or the mapping degree) and can use examples to illustrate concepts and theorems, e.g. they can compute the homology of space more complicated that projective spaces.

Furthermore, the students recognise further-reaching connections to mathematical facts already acquired. They can transfer and apply the knowledge and methods they have learnt so far to deeper mathematical problem areas. Students also expand their mathematical intuition as a result of more intensive study.
In the tutorials, students develop their ability to discuss mathematical topics and thus further prepare themselves for the requirements of the Master's module, in particular for the scientific discussion within the Master's seminar presentation and the defence of their Master's thesis.

The following basic content of teaching from the field of Algebraic Topology is compulsory:

• Singular and cellular homology
• Cell complexes and simplicial sets
• Classical applications, e.g. invariance of dimension, invariance of domain, hairy ball theorem, Brouwer´s fixed point theorem, Jordan separation theorem, theorem of Borsuk-Ulam
• Computations of relevant examples, e.g. Stiefel and Grassmann manifolds or orthogonal/unitary groups
• Fundamental classes and mapping degrees

In addition, the following content of teaching can be covered, for example:

• Simplicial homology and simplicial approximation
• Singular and cellular cohomology
• Homology of products
• Ext, Tor and the coefficient theorems
• Dold-Kan correspondence
• de Rham cohomology
• Sheaf cohomology
• Euler’s polyhedral formula
• Lefschetz fixed point formula
• Planarity of graphs
• Cellular approximation

Topics from a bachelor programme, such as topological spaces and fundamental groups, linear algebra over integers are required, basic knowledge of chain complexes and homology is recommended

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