Math 636 Spring 2019

Instructor: Robert Lipshitz

Basic course information

Time: MWF 3:00-3:50 p.m.
Place: Deady 104
Textbook: Allen Hatcher, Algebraic Topology. Available for download here.
Office hours: M 12-1, F 1:45-2:45. Subject to change.
Teaching assistant: Dan Raies
TA office hourse: TBD
Final exam: per Registrar's schedule.


Math 635.

Description and goals

The third quarter of the algebraic topology sequence focuses on properties, computations, and applications of higher homotopy groups, as well as their relations to homology and cohomology.

Particular "learning objects" for the course include:

  • Be able to compute certain homotopy groups using fibrations, the Hurewicz isomorphism theorem, and the Freudenthal suspension theorem.
  • Understand and apply Whitehead's theorem.
  • Understand the relationship between Eilenberg-MacLane spaces and cohomology.
  • Understand an apply the basics of obstruction theory.



Homework 35%
Midterm 25%
Final exam 40%


The midterm will be a closed-note, closed-book, 3-hour take-home exam. The final will be a 2-hour exam during in the assigned final exam slot.


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.
  • 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 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 4/1

More on intersection product.

Bredon, VI.11

More on intersection product.


Topology on function spaces. Homotopy revisited. Path space and loop space.

pp. 529-532, +
2 4/8

Homotopy groups. Relative homotopy groups.

This is also the last day to drop the class without a W.

pp. 339-346, 421-426

Whitehead's theorem.


Cellular approximation.

pp. 348-357
3 4/15

CW models


More on CW models


Freudenthal suspension theorem.

pp. 360-366
4 4/22

Hurewicz isomorphism theorem.

pp. 366-375, +

Proof of Hurewicz theorem.


More Hurewicz.

5 4/29

Fiber bundles and fibrations.

pp. 375-384

Long exact sequence for a fibration.


Hopf invariant.

pp. 427-428
6 5/6

Review / catch-up. (Midterm exam distributed.)


More examples of fibrations. Whitehead products.


Stable homotopy groups.

pp. 384-388
7 5/13

Eilenerg-MacLane spaces and cohomology

pp. 393-405



More on fibrations

The last day to withdraw from the class is 5/19.

pp. 405-409
8 5/20

Postnikov towers.

pp. 410-415

More Postnikov towers.


Obstruction theory.

pp. 415-419
9 5/27

Memorial day - no class.


More obstruction theory, application of obstruction theory.


Leray-Hirsch theorem.

10 6/3

Brown representability theorem.

4.E, 4.F

Dold-Thom theorem.


Steenrod algebra.




  • 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.