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401-4571-22L 4 Credits BSC , MSC D-MATH

Topology of Manifolds

Lecturers & Examiners: Dr. Danica Cekic
VVZ CR n/a

Last Updated: 2026-02-05 16:37:24

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.

Resources

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.

General Information

Language
English
Levels
BSC , MSC

Examination

Type
end-of-semester examination
Mode
written 90 minutes
Aids
none

Course Components

Type Title Time & Place Hours
lecture Topology of Manifolds
  • Tue 08:15-10:00 (HG F 26.5)
2 h weekly

Offered In