Fall 2023
Course Information
| Course | MATH 4565 · Topology |
| Instructor | Alex Suciu |
| Course Web Site | https://suciu.sites.northeastern.edu/courses/math4565-fall2023 |
| Time and Place | 408 Ell Hall, Tue, Fr 9:50 am–11:30 am |
| Office | 435 LA – Lake Hall |
| a.suciu@northeastern.edu | |
| Office Hours | Wednesday 10:30 am–11:30 am & Friday 1:45 pm–2:45 pm, or by appointment |
| Prerequisites: | MATH 3150 – Real Analysis |
| Textbook | Introduction to Topological Manifolds, Second Edition, by John M. Lee, Graduate Texts in Mathematics, vol. 202, Springer, 2011. |
| Additional books | Topology (2nd Edition), by James R. Munkres, Pearson, 2018. Algebraic Topology, by Allen Hatcher, Cambridge University Press, 2001 (pdf file). |
| Grade | Based on problem sets (50%), midterm exam (20%), and final exam (30%). It is expected that you will work on the problem sets together; however, they must be written up separately. |
Course Description
This course provides an introduction to the concepts and methods of Topology. It consists of three, inter-connected parts.
1. Topological Spaces and Continuous Maps
This part of the course serves as an introduction to General Topology. The objects of study are topological spaces and continuous maps between them. Key is the notion of homeomorphism, which leads to the study of topological invariants. The main properties that are studied are connectedness, path connectedness, and compactness. We also introduce several constructions of spaces, including identification spaces, and discuss topological manifolds and topological groups.
2. Fundamental Group and Covering Spaces
This part of the course is a brief introduction to Geometric Topology. It starts with the definition of the fundamental group of a space, and various methods to compute it, such as the Seifert-van Kampen theorem. It proceeds with the classification of surfaces, and an introduction to the theory of covering spaces.
3. Simplicial Complexes and Simplicial Homology
(Time permitting) This a brief introduction to the methods of Combinatorial Topology and Homological Algebra, and serves as an advertisement for some of the recent advances in Computational Topology and Topological Data Analysis. It starts with simplicial complexes and their realizations and proceeds to simplicial homology groups, and ways to compute them. Time permitting, we will illustrate these techniques with concrete examples, and derive some applications.
Assignments and Exams
- Assignment #1 (due September 22): Exercise 2.4 (a),(b),(c) · Exercises 2.21 & 2.23 · Problem 2-2 · Problem 2-4 · Problem 2-5.
- Assignment #2 (due October 4): Exercise 2.32 · Exercise 2.35 · Problem 2-8 · Problem 2-10 · Problem 2-11 · Exercise 3.7.
- Assignment #3 (due October 18): Exercise 3.45 · Problem 3.2 · Problem 3.3 · Problem 3.6 · Problem 4.1 · Problem 4.12
- Midterm Exam: due November 2.
- Assignment #4: due November 16.
- Final Exam: due December 13.
Handouts
- Handout 1: Inverse images and direct images
- Handout 2: Interior, closure, and boundary
- Handout 3: Homotopy
For more information (including handouts and exams) see these older syllabi: Spring 2006 · Spring 2008 · Spring 2010 · Spring 2016 · Spring 2018 · Fall 2021 · Fall 2022.