General Prerequisites:
Part A Topology is essential and Group Theory is recommended.
Course Term: Michaelmas
Course Lecture Information: 16 lectures
Course Weight: 1
Course Level: H
Assessment Type: Written Examination
Course Overview:
This course introduces the important link between topology and group theory. On the one hand, associated to each space, there is a group, known as its fundamental group. This can be used to solve topological problems using algebraic methods. On the other hand, many results about groups are best proved and understood using topology. For example, presentations of groups, where the group is defined using generators and relations, have a topological interpretation. One of the highlights of the course is the Nielsen-Schreier Theorem, an important, purely algebraic result, which is proved using topological techniques.
Learning Outcomes:
Students will develop a sound understanding of simplicial complexes, cell complexes and their fundamental groups. They will be able to use algebraic methods to analyse topological spaces and compute the fundamental groups of many spaces, including compact surfaces. They will also be able to address questions about groups using topological techniques.
Course Synopsis:
Homotopic mappings, homotopy equivalence. Simplicial complexes. Simplicial approximation theorem.
The fundamental group of a space. The fundamental group of a circle. Application: the fundamental theorem of algebra. The fundamental groups of spheres.
Free groups. Existence and uniqueness of reduced representatives of group elements. The fundamental group of a graph.
Groups defined by generators and relations (with examples). Tietze transformations.
The free product of two groups. Amalgamated free products.
The Seifert-van Kampen Theorem.
Cell complexes. The fundamental group of a cell complex (with examples). The realization of any finitely presented group as the fundamental group of a finite cell complex.
Covering spaces. Liftings of paths and homotopies. A covering map induces an injection between fundamental groups. The use of covering spaces to determine fundamental groups: the circle again, and real projective \(n\)-space. The correspondence between covering spaces and subgroups of the fundamental group. Regular covering spaces and normal subgroups.
Cayley graphs of a group. The relationship between the universal cover of a cell complex, and the Cayley graph of its fundamental group. The Cayley 2-complex of a group.
The Nielsen-Schreier Theorem (every subgroup of a finitely generated free group is free) proved using covering spaces.