Professor Alexandru I. Suciu
MATH 4565 · Topology
Fall 2021

* Course Information

Course MATH 4565 · Topology: CRN 18969, Section 1
Instructor Alex Suciu
Course Web Site web.northeastern.edu/suciu/MATH4565/utop.fa21.html
Time and Place Tue, Fr 9:50am–11:30am in 460 Ryder Hall
Office 435 LA – Lake Hall
Phone (617) 373-3833
Email a.suciu@northeastern.edu
Office Hours Tue 1:15pm-2:15pm, Fr 11:45am-12:45pm 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
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
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

  • Homework 1 (due September 28):  Exercises 2.21 & 2.23; Exercises 2.29 & 2.32; Problem 2-1 (a),(c),(e); Problem 2-2; Problem 2-4; Problem 2-5.
  • Homework 2 (due October 8):  Exercise 2.35; Problem 2-8; Problem 2-10; Problem 2-11; Exercise 3.6; Exercise 3.7.
  • Homework 3 (due October 19):  Exercise 3.34 (for real-valued maps); Exercises 3.43, part (c) and Exercise 3.45; Exercise 3.55; Exercise 3.62, parts (a), (b), (c); Problem 3-5 (for two maps); Problem 3-6.
  • Homework 4 (due November 16):  Exercise 4.38; Exercises 4.78 and 4.79; Problem 4-7; Problem 4-23, parts (a), (b), (d); Problem 4-27
  • Homework 5 (due December 3):  Exercise 7.14, Problem 7.1, Problem 7.2, Problem 7.3, Problem 7.7, Problem 7.12.

* Handouts


* Exams


Home Started: April 16, 2021
Last modified: December 7, 2021
web.northeastern.edu/suciu/MATH4565/utop.fa21.html