# 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

• Instructor: Robert Young (ryoung@cims.nyu.edu)
• Office: WWH 601
• Office hours: Wednesdays, 4-5, in WWH 601. Check Brightspace or email me for a Zoom link.
• Lectures: WWH 517, Thursdays 5:10-7:10
• Suggested texts:

 Assignments 30% Midterm 25% Final exam 45%

• Midterm: TBA
• Final: TBA

## 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