DescriptionThe fundamental group of a topological space gives a beautiful and far reaching link between Topology and Group Theory. In particular, it allows us to study groups using topological spaces. In order to do this effectively for a given group G, one needs a 'good' space X whose fundamental group is G. A first, slightly surprising result in this direction is that for every group G there exists a space whose fundamental group is G. The problem with this result is that the general procedure used may not lead to a space one would consider 'good'.Of course, the term 'good topological space' is not well defined and can have different meanings in different situations. One interpretation that we are going to investigate in this project leads to the notion of finiteness properties of groups. A simple question in this direction is whether for a given group G there exists a finite simplicial complex whose fundamental group is G or at least surjects on G. The latter condition means that G is finitely generated, a condition you may have seen in the Algebra course. The first condition means that G is finitely presented, a desirable condition that is not satisfied by every finitely generated group, and which can also be expressed in completely algebraic terms. These concepts were generalised by Wall in the sixties in his work on understanding the homotopy type of finite CW complexes (a class of topological spaces more flexible than simplicial complexes, but still very well behaved) and have an important impact in both group theory and topology. We will learn about these concepts and their subtle differences in this project. One interesting feature is that subgroups of finitely presented groups need not be finitely generated or finitely presented themselves. While these are purely group theoretic questions, they are best studied in connection with topology. In particular, covering space theory, a fundamental group analogue of Galois Theory, is a particularly useful tool here. PrerequisitesTopology III.ResourcesA good source for basics on finitely generated and presented groups is the book of Lyndon and Schupp Combinatorial Group Theory. The book of Geoghegan Topological Methods in Group Theory is a good source for the finiteness properties.Some slightly more advanced articles which are here and here. |
email: D Schuetz