DescriptionAn n-dimensional manifold is a topological space that is locally like standard n-dimensional Euclidean space R^n. However, the local patches can be glued together so that a non-trivial topology arises. In this project, focus will be given on 3-dimensional manifolds. One can think of these as possible models for a (spatial) universe in which we live. Lens spaces are a big class of examples of such manifolds for which there is a complete classification.Algebraic topology provides certain abelian groups, the singular homology groups, which are associated to a manifold. These allow to say, for instance, that two given manifolds cannot be the same (up to a suitable notion of isomorphism). For lens spaces, this distinction is not fine enough. There is another invariant of manifolds, called the Reidemeister torsion , which is a more subtle invariant. It provides a complete classification of Lens spaces. In this project you will learn about 3-dimensional manifolds and ways of constructing these - by means of glueing together manifolds with boundary along their common boundary, or by a process called Dehn surgery. You will study chain complexes and Reidemeister torsion, and derive the classification of Lens spaces. You will also see that lens spaces arise on Dehn surgery on knots: For the unknot this is easy to see, but there are more surprising examples of non-trivial knots which have a Dehn surgery resulting in a Lens space. PrerequisitesGeometric Topology III, Algebraic Topology IV (it is sufficient to take the class along with the project).ResourcesThe following Wikipedia pages give a concise picture, but without prerequisits that you will learn in the project they might only give a very rough idea:The following YouTube-video describes Dehn surgery visually: Relevant books/lecture notes include the following:
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email: Andrew Lobb email: Raphael Zentner