Math 635 Winter 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: Monday 10:20-10:50, 12:00-12:30; Friday 12:00-12:30; and by appointment. By Zoom. Subject to change.
Teaching assistant: Wei Zhang.
Final exam: take home.


Math 634.

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 continues where Math 634 left off, studying properties, computations, and applications of singular homology and cohomology. The second half of the course focuses on the (co)homology of manifolds and various forms of Poincaré duality, as well as further computations and applications.

Particular "learning objects" for the course include:

  • Being able to compute and apply homology and cohomology, including the ring structure on cohomology.
  • Understanding and applying PoincarĂ©, Lefschetz, and Alexander duality for the (co)homology of manifolds.



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.
  • 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 1/4

Review: singular chain complex, functoriality, homotopy invariance, reduced homology, long exact sequence, excision.

pp. 102-119.

Proof of excision.

pp. 119-126.

Mayer-Vietoris sequence

pp. 149-153
2 1/11

Degree, invariance of domain, Jordan curve theorem

pp. 134-137, 169-173.

Cellular homology.

pp. 137-146.

More on cellular homology. 1-dimensional Hurewicz theorem for CW complexes.

3 1/18

Martin Luther King day (no class)


Axioms for homology.


Simplicial approximation, Lefschetz fixed point theorem.

4 1/25

Catching up. Simplicial, singular, and cellular cohomology over ℤ. (Also, aside about split exact sequences.)

pp. 185-189.

Homology and cohomology with arbitrary coefficients. Euler characteristic.

pp. 146-147, 153-155, 197-204.

Basic properties of cohomology (homotopy invariance, long exact sequences, etc.)

5 2/1

Universal coefficient theorem for cohomology.

Midterm exam distributed.

pp. 190-197.

Universal coefficient theorem homology.

pp. 261-267.

Cup product.

Midterm exam due.

pp. 206-210.
6 2/8

More on cohomology ring.

pp. 211-217.

More on cohomology ring.


More on cohomology ring.

7 2/15

Cross product

pp. 210-211.

Kunneth theorems

pp. 218-223, 268-280, +
or: 214-219, 268-280, +
(depending on your version of Hatcher)

More on Kunneth theorems.

8 2/22

Poincaré duality: motivation. Cohomology of projective spaces. Orientations.

pp. 230-241

More on orientations.


More on orientations. Compactly supported cohomology.

pp. 242-252
9 3/1

Poincaré duality


Lefschetz duality

pp. 252-254.

Alexander duality

pp. 254-257.
10 3/8

Smooth manifolds, transversality


Vector bundles, Thom isomorphism theorem


Intersection product