Topology I, Fall 2021

Topology is the study of topological spaces, continuous maps between them, and properties preserved under continuous transformations. Topological spaces are ubiquitous in mathematics; Banach spaces, metric spaces, and varieties are all types of topological spaces.

In this course, we will start by introducing metric and general topological spaces, including basic properties like compactness and connectedness. We will then study elements of algebraic topology, including fundamental groups and covering spaces, homotopy and the degree of maps.

Basic information

Grading scheme

Final exam45%

Exam dates

Problem sets

Late problem sets will not be accepted except in the case of an emergency. At the end of the semester, your lowest problem set grade will be dropped from your average. This is meant to accommodate non-emergency absences, so try not to use this unless you have to.

Course outline

09/02 Introduction, metric spaces and topological spaces Munkres, §12, §20 Notes
09/09 Examples of topological spaces Munkres, §13-16 Notes
09/16 Continuous functions Munkres, §17-18 Notes
09/23 Connectedness Munkres, §23-24 Notes
09/30 Compactness Munkres, §26-27 Notes
10/7 Manifolds and complexes Hatcher, §0 Notes
10/14 The fundamental group Hatcher, §1.1 Notes
10/21 The fundamental group of the circle Hatcher, §1.1 Notes
10/28 Deformation retracts and homotopy equivalences Hatcher, §0, §1.1 Notes
11/04--11/11 The van Kampen Theorem Hatcher, §1.2 Notes (pages 1-8)
11/18 Covering spaces Hatcher, §1.3 Notes (pages 9-12)
12/02-- Classification of covering spaces Hatcher, §1.3 Notes