Math 636 Spring 2021

Instructor: Robert Lipshitz

Basic course information

Time: MWF 11:00-11:55 a.m.
Place: Zoom. Zoom link posted in Canvas.
Textbook: Allen Hatcher, Algebraic Topology. Available for download here.
Office hours: TBD and by appointment. By Zoom. Subject to change.
Teaching assistant: Wei Zhang.
Final exam: take home.


Math 635.

Resources for this online-synchronous class

We will use:

  • Zoom for the lectures and office hours. I prefer, but do not require, that you have your video camera on during lecture.
  • A OneNote "class notebook" as a second place you will have access to what I write during lecture. You will receive a link to the notebook by e-mail.
  • Canvas for announcements, submitting homework, seeing feedback on homework, posting solutions, and tracking scores.

Description and goals

This course is focused mainly on homotopy groups of spaces and their applications, particularly to what Hatcher calls the homotopy interpretation of cohomology.

Particular "learning objects" for the course include:

  • Being able to compute homotopy groups in special cases, using the Hurewicz theorem, long exact sequence for a fibration, and other elementary techniques.
  • Understanding and applying Whitehead's theorem to show when spaces are homotopy equivalent.
  • Reinterpreting cohomology as a representable functor, and understanding how this classifies cohomology operations.



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:

  • Glen Bredon, Topology and Geometry.
  • William Massey, Algebraic Topology: An Introduction and A Basic Course in Algebraic Topology.
  • John McCleary, A User's Guide to Spectral Sequences.
  • 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 11:59 p.m. on Fridays most weeks. 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




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

Week Date Topic Textbook sections
1 3/29

More on intersection product.

Bredon, VI.11

More on intersection product.


More on intersection product.

Saturday is last day to drop class without a W.

2 4/5

Homotopy groups. Relative homotopy groups.

pp. 339-346, 421-426

Whitehead's theorem.


Cellular approximation.

pp. 348-357
3 4/12

CW models


More on CW models


Freudenthal suspension theorem.

pp. 360-366
4 4/19

More on Freudenthal suspension theorem and its proof.


Hurewicz isomorphism theorem.

pp. 366-375, +

More Hurewicz.

5 4/26

Fiber bundles and fibrations.

pp. 375-384

Long exact sequence for a fibration.


Hopf invariant.

pp. 427-428
6 5/3

Topology on function spaces. Homotopy revisited.

pp. 529-532, +

More examples of fibrations.


Stable homotopy groups.

pp. 384-388
7 5/10

Eilenberg-MacLane spaces and cohomology.

pp. 393-405

More on Eilenberg-MacLane spaces and cohomology.


More on fibrations.

The last day to withdraw from the class is 5/16.

pp. 405-409
8 5/17

Postnikov towers.

pp. 410-415

More Postnikov towers.


Filtered complexes and spectral sequences

9 5/24

Serre spectral sequence: statement, first examples


Sketch of proof of Serre spectral sequence


Statement and proof of Serre's mod-C Hurewicz theorem

10 5/31

Memorial day - no class.


Applications of the mod-C Hurewicz theorem


Cohomology of Eilenberg-MacLane spaces and the Steenrod algebra