Math 634 Fall 2018

Instructor: Robert Lipshitz

Basic course information

Time: MWF 9:00–9:50 p.m.
Place: 209 Deady Hall.
Textbook: Allen Hatcher, Algebraic Topology. Available for download here.
Office hours: Monday 12-1, Friday 2-3. Subject to change.
Teaching assistant: Bradley Burdick
TA office hourse: Monday 2-3, Friday 10-11. Subject to change. Monday OH are in Deady 209.
Final exam: per Registrar's schedule.


This course requires a firm understanding of the material from Math 531 (point-set topology) and Math 544 and 545 (abstract algebra, particularly groups, rings, and modules). This course is aimed at graduate students in the Mathematics Department, who should decide whether to take this or an alternative in consultation with their departmental advisors. Anyone else must obtain the instructor's permission to take this course.

Description and goals

This is the first quarter of the 600-level sequence in topology, and discusses the fundamental group and covering spaces and the definitions of singular homology and cohomology. These are basic tools in modern topology, and these tools and related ideas are also important in differential geometry, algebraic geometry, and other areas. While the course will include some applications, more are given in the second quarter. The course is structured to make it possible for students to understand the Spectral Sequences course in the winter if they want to.

Particular "learning objects" for the course include:

  • Work with the notions of homotopy and homotopy equivalence, and be able to prove maps are homotopic and spaces are homotopy equivalent.
  • Be able to construct and prove properties of CW complexes.
  • Understand and be able to compute the fundamental group and use it for topological applications.
  • Understand and apply the definition, fundamental lifting theorem, and classification theorem for covering spaces.
  • Be able to define and compute simplicial and singular homology and cohomology and prove their basic properties.

Provisional course webpages for the next two courses can be found here: Math 635; Math 636.



Homework 35%
Midterm 25%
Final exam 40%



The textbook for the class is Algebraic Topology by Allen Hatcher. It is available online or in print. Some other useful textbooks include:

  • William Massey, Algebraic Topology: An Introduction and A Basic Course in Algebraic Topology.
  • James Munkres, Elements of Algebraic Topology and Topology.
  • Edwin Spanier, Algebraic Topology.
  • J. Peter May, A Concise Course in Algebraic Topology. Apparently available for download from Peter May's website.
  • Anatoly Fomenko and Dmitry Fuchs, Homotopical Topology. Available for download via SpringerLink, through the library.

Students are expected to read the sections in the textbook before coming to class each day. The relevant sections are listed in the syllabus below.


Homework is due at the beginning of class on Wednesdays most weeks. The first homework assignment is due on Friday of the first week. You may discuss the problems with other students, but must write up your solutions to the problems by yourself. Any resources you use other than the textbook must be cited in your homework. You may not use electronic resources (e.g., Google) other than the textbook and recommended textbook. Failure to follow this policy constitutes cheating; if you are caught cheating on the homework you will receive a 0 for the homework portion of the class and will be reported to the administration.

Students with disabilities

I, and the University of Oregon in general, are committed to an inclusive learning environment. If you have a disability which may impact your performance on exams, please contact the Accessible Education Center to discuss appropriate accommodations. If there are other disability-related barriers to your participation in the course, please either discuss them with me directly or consult with the Accessible Education Center.

Assignments, handouts, and other resources



LaTeX resources

  • TeXlive. This is the LaTeX distribution I have found easiest to work with. It's available for Linux, Mac, and Windows. The Mac version goes under the name MacTeX. This also installs TeXworks (Windows) and TeXshop (Mac); if you're new to using LaTeX, these are probably good editors to start with. TeXworks is also available for Mac and Linux.
  • A (Not So) Short Introduction to LaTeX2ε. This is the document I learned TeX from.



Sections refer to Hatcher's book. + denotes material beyond that covered in these sections of the textbook.

Week Date Topic Textbook sections
1 9/24

Course overview. Finite CW complexes.

pp. 5-8, 519-529

More examples of CW complexes. Infinite CW complexes.


More on topology of infinite CW complexes. Examples.

Homework 1 due.

The last day to drop the class without a W is 9/29.

2 10/1

New spaces from old.

pp. 8-10

Homotopy, deformation retractions, homotopy equivalence. Homework 2 due.

pp. 1-4

Constructing homotopy equivalences.

pp. 10-17
3 10/8

Path homotopy and the fundamental group.


The fundamental group of the circle and spheres. Homework 3 due.


Functoriality under continuous maps, change of basepoint. Brouwer fixed point theorem.

4 10/15

Van Kampen theorem: statement, examples.


Van Kampen theorem: proof. Homework 4 due.


Fundamental groups of CW complexes

5 10/22

Covering spaces: definition, examples.

§1.3, pp. 56-63

Lifting properties of covering spaces. Homework 5 due.


Deck transformations.

pp. 70-76
6 10/29 Midterm exam.  

Construction of the universal cover. Homework 6 due.

pp. 63-70, +

Classification of covering spaces.

7 11/5

Simplices, simplicial complexes.

pp. 97-107, +

Simplicial homology and cohomology. Homework 7 due.


Functoriality of simplicial homology and cohomology.

The last day to withdraw from the class is 11/11.

8 11/12

Singular homology and cohomology: definition.

pp. 108-110, +

Homotopy invariance of singular (co)homology. Homework 8 due.

pp. 110-113, +

Relative (co)homology and excision

pp. 113-128
9 11/19

Snake lemma


Long exact sequence of a pair. Homework 9 due.

  11/23 Thanksgiving break -- no class
10 11/26

Mayer-Vietoris sequences.

pp. 149-153

Cup product on cohomology. Homework 10 due.

pp. 206-218

More computations and examples.




  • Read the sections in the textbook before class, and again after class. Read with a pencil and paper in hand. Note down points that confuse you, and come back to them later to make sure you understand them. Work through the examples. When the author says something is "clear" that means it is not obvious -- make sure you understand why it is true.
  • Read the suggested sections in the secondary textbook, either before or after class (or both).
  • Get help as soon as you are confused. Your best options for help are my office hours, other students in the class, and other students not in the class.
  • Start on the homework problems as soon as we have covered the corresponding material in class.
  • Start working on the homework by yourself. After you have spent at least half an hour on each problem, try describing your solutions or where you are stuck to other students in the class. (Discussing the problems is helpful in both cases.) Then, go back and write (or type) your solutions nicely by yourself. If you work in a group, it is easy to think you've learned the material when you have not.
  • Do not look for hints or solutions on the internet. I guarantee you will not learn the material that way, even if you feel like you do.
  • Solve extra problems.
  • Think about the material, and how different parts of the material relate to each other, and examples of the different concepts, and counter-examples if you drop hypotheses from theorems, and how the material relates to material from other classes, constantly -- whenever there is a pause in your day.