|Abstract||This will be an introduction to geometric topology, a field of mathematics concerned with topological properties of manifolds. We will study both topological and smooth manifolds, and prove some fundamental results about them (like the Schoenflies theorem, the generalised Poincaré conjecture, the existence of exotic smooth structures), several of which have been awarded with Fields medals.|
|Objective||At the end of the course students will be able to differentiate between three types of manifolds, give examples showing various phenomena, and prove some classical results. They will understand what kinds of arguments are used in each of the cases, and where the difficulties arise. Moreover, they will become familiar with many open problems that are guiding current research, especially in the peculiar dimension four.|
|Content||There are several notions of a manifold -- namely, topological, piecewise-linear, and smooth -- and only in 1956 did it become clear that these objects are in fact distinct, thanks to the construction by J. Milnor of multiple smooth structures on a single topological manifold. In this course we will start with basic definitions and properties of the three types of manifolds, building our way up to cover some fundamental results.|
We will first study handle decompositions, transversality and the Whitney trick, the s-cobordism theorem, the Poincaré conjecture, and the Schoenflies theorem. Possible further topics include torus tricks, smoothing theory, exotic spheres, the Rohlin theorem, exotic 4-manifolds.
|Literature||• See the lecture notes and a reference list at https://maths.dur.ac.uk/users/mark.a.powell/topological-manifolds.html|
• Hirsch, M. Differential topology. Graduate Texts in Mathematics, No. 33. Springer-Verlag, New York-Heidelberg, 1976.
• Kosinski, A. Differential manifolds. Pure and Applied Mathematics, 138. Academic Press, Inc., Boston, MA, 1993.
• Scorpan, A. The wild world of 4-manifolds. American Mathematical Society, Providence, RI, 2005.
|Prerequisites / Notice||We will assume familiarity with point-set topology, the fundamental group (as covered in the course Topology), homology (as covered in Algebraic Topology I), and some basics of differential topology and vector bundles (as covered in Differential Geometry I). Some familiarity with cohomology and Poincaré duality would be useful.|