Math 671: Topology I - Course Syllabus

Course: Math 671 Topology I
Credits: 3 credits, in-person
Meeting Time: TuTh 10–11:15 in LGRT 1322

Instructor Details

Instructor: Paul Gunnells
Office and Phone: LGRT 1115L, 5–6009
Email: gunnells@umass.edu
Office Hours: TBA

Course Description

This course, together with Math 672, presents topics in point-set topology and algebraic topology. In the first semester, the emphasis is on point-set topology and the beginnings of algebraic topology (specifically the fundamental group). The second semester treats homology and cohomology.

The topics this semester are:

Part I: Basic point-set topology, constructions of topological spaces, connectedness, compactness, countability and separation axioms, CW complexes and topological manifolds.
Part II: Introduction to algebraic topology, cell complexes, homotopy, fundamental group, covering spaces.

Prerequisites

Undergraduate real analysis and group theory, and a strong facility with proofs.

Textbooks

(Part I) Lee, John M. Introduction to Topological Manifolds. 2nd ed. Graduate Texts in Mathematics 202. New York: Springer, 2011.
(Part II) Hatcher, Allen. Algebraic Topology. Cambridge: Cambridge University Press, 2002. Available free here.

Additional resources. These are not required but may be helpful.

Munkres, James R. Topology. 2nd ed. Upper Saddle River, NJ: Prentice Hall, 2000.
Massey, William S. Algebraic Topology: An Introduction. New York: Springer-Verlag, 1977.
Miller, Haynes. Lecture notes on algebraic topology. Available free here

Course Objectives

During this course, students will practice and develop greater facility with reading, analyzing, producing and writing complex, structured proofs. By the end of the course, students should:

Understand and be able to use different constructions of a topology on a set: the metric topology, the topology generated by a basis, the product, coproduct, subspace, and quotient topologies;
Be able to work with connectedness, path-connectedness, and compactness as well as their local variants; know consequences of these properties as well as ways to produce spaces having them;
Know the basic topological properties of CW complexes; be able to construct a CW structure on some familiar spaces;
Be able to compute examples of the fundamental group using Seifert–Van Kampen or CW structures; be able to use pi_1 both as a group and as a functor to make inferences about spaces and their homotopy type;
Understand the Galois correspondence between subgroups and covering spaces, and be able to use it to study the structure of covering maps and lifting problems.

Grading Scheme

Your course grade will be based on homework, an in-class exam, a final exam, and class participation.

Component	Weight	Description
Homework	40%	Assigned approximately weekly.
In–Class Exam	20%	45–minute exam on definitions, theorem statements, and key examples (tentatively Thursday, Nov 19).
Final Exam	30%	Cumulative exam
Participation	10%	Based on asking questions, attending office hours, or showing active interest.

Note that the date and time of the final exam is set by the University. Exams will not be given earlier.

Grading Scale

Grade	Percentage
A	90–100%
A–	86–90%
B+	82–86%
B	78–82%
B–	74–78%
C+	70–74%
C	66–70%
F	0–54%

According to a department policy, final numerical scores used in computing final grades are truncated down to the nearest integer, and are not rounded.

Homework Rules and Guidelines

I encourage students to discuss the homework problems with each other and to work in groups, but if you do so, you must list the names of all people you worked with, and you must write up your solutions completely independently.

Solutions to homework problems must be handwritten. Solutions compiled from LaTeX or other electronic document formats (Word, OpenOffice, etc) are not allowed and will not be accepted.

You will hand in your homework as pdf on Gradescope. You may use scanners in the library or a scanning app to convert your handwritten sheets to pdf.

Homework must be submitted on time. Late assignments will not be accepted. At the end of the term some homework scores will be dropped before computing your homework contribution to the final grade.

Course Schedule

This is tentative and may be changed.

Week	Dates	Topics Covered
Week 1	9/8, 9/10	Introduction, topologies, open and closed sets
Week 2	9/15, 9/17	Bases for a topology, defining manifolds, subspace topology
Week 3	9/22, 9/24	Topologies for products, disjoint unions, quotients
Week 4	9/29, 10/1	Gluing spaces, Connectedness and path–connectedness
Week 5	10/6, 10/8	Compactness, the closed map lemma
Week 6	10/13, 10/15	Local compactness, paracompactness and partitions of unity, embeddings and Urysohn's Lemma
Week 7	10/20, 10/22	Proper maps. CW complexes
Week 8	10/27, 10/29	Starting algebraic topology: homotopy
Week 9	11/3 (No class), 11/5	The fundamental group, fundamental group of the circle, induced homomorphisms
Week 10	11/10, 11/12	Brouwer f.p. theorem, FTA, Seifert–van Kampen theorem.
Week 11	11/17, 11/19	Seifert–van Kampen theorem continued, Covering spaces In–class exam: 11/19.
Week 12	11/24 (No class)	Thanksgiving recess
Week 13	12/1, 12/3	Classification of surfaces
Week 14	12/8, 12/10	Galois correspondence on covering spaces; deck transformations
Week 15	12/15	Review for exam
Final Exam	Tues 12/22	Final Exam (10:30 AM – 12:30 PM), LGRT 1322

University Policies

For official University policies regarding Accommodations, Academic Integrity, and Title IX, please see here.