Fall 2025
Instructor: Marc Rieffel
Lectures: MWF 12:10-1:00 pm, Room 219 Dwinelle
Course Control Number: 23827
Office: 811 Evans
Office Hours: TBA
Prerequisites: Math 104 and 113 or equivalent.
Required Text: Topology, 2nd Ed., J. R. Munkres, Prentice Hall, 2000 (paperback version is ok, and much less expensive).
Syllabus: Topology is the mathematics of continuity. It considers how the points of "spaces" hang together, while ignoring distances. (Distances can be considered the concern of "geometry", as studied in Math 140.). Thus the surface of a donut is topologically "the same" as the surface of a one-handled teacup. General relativity tells us that distances in our universe change with time, as masses warp space while they move. But maybe in a global way the topology of our universe does not change. Topology is important for virtualy every branch of mathematics and its applications. In the early weeks of the course we will discuss how to define "topological spaces" so that all of this has a precise meaning. Since all of this strongly involves the ideas of continuity, the material of Math 104 will be very important.
How then do we actually prove such things as "the surface of a donut is topologically not the same as the surface of a ball", as well as many less intuitive facts? Algebraic topology provides the main tools for handling this and many related questions. These tools consist of various methods for systematically attaching algebraic objects, such as groups and rings and the homomorphisms between them, to topological spaces and the continuous functions between them. (Thus the material of Math 113 will be very important.) Then one can compute with these algebraic objects to try to answer the topological questions. In the latter part of the course we will consider some elementary versions of these tools, and we will apply them to surfaces and other interesting spaces.
In my lectures I will try to give well-motivated careful presentations of the material, with interesting examples. I encourage class discussion.
Grading: The final examination is projected to take place on Friday December 19, 2025, 11:30 AM - 2:30 PM. The final examination will count for 50% of the course grade. There will be no early or make-up final examination. There will be two midterm examinations, at the regular class time, on dates to be decided very early in the semester. Each midterm exam will count for 20% of the course grade. Make-up midterm exams will not be given; instead, if you tell me ahead of time that you must miss a midterm exam, then the final exam and the other midterm exam will count more to make up for it. If you miss a midterm exam but do not tell me ahead of time, then you will need to bring me a very persuasive doctor's note or equivalent in order to try to avoid a score of 0. If you miss both midterm exams the circumstances will need to be truly extraordinary to avoid a score of 0 on at least one of them.
Homework: Homework will be assigned at nearly every class meeting, and be due before the following class meeting. Homework will count for 10% of the course grade. Students are strongly encouraged to discuss the homework and the course content with each other, but each student should write up their own solutions, reflecting their own understanding, to turn in. Even more, if students collaborate in working out solutions, or get specific help from others, they should explicitly acknowledge this help in the written work they turn in. This is general scholarly best practice. There is no penalty for acknowledging such collaboration or help. Similarly, some of the homework problems may have solutions in various books or papers or other sources such as A. I.. I strongly recommend that students try to solve the problems without looking at such sources. But if such sources are used, then again, scholarly best practice is to cite such sources. Again, there is no penalty for doing this.
We will probably use GradeScope for posting the homework assignments and for students to upload their solutions.
Accomodations: Students who need special accommodations for examinations should make sure I receive the appropriate paperwork, and must tell me between one and two weeks in advance of each examination what specific accommodation they need for that exam, so that I and the department will have enough time to arrange it.
Using TEX: Some students may find it useful to write up some of their homework solutions in TEX, more specifically LATEX. (But I do not at all require this.) LATEX is a powerful mathematical typesetting program which is widely used in the sciences, engineering, etc., for documents that use a lot of mathematical symbolism. Thus learning to use TEX is a valuable skill if you work in such fields.
You can freely download versions of TEX onto your computer.
Several guides to using TEX are listed on the department's computer support web pages.
Others can be found by searching on-line for "latex tutorial'' or "latex manual''.
The best way to start learning TEX is not by trying to compose the long header
that is needed, but rather by
having a file that already has a header, and then gradually modifying that file
as you learn how TEX works.
Since the output from TEX is a PDF file that can be directly uploaded to GradeScope,
this would eliminate the need to scan pages of paper for uploading to GradeScope.
Changes:
The above procedures are subject to change.
This page was last updated on 07/20/2025
If you use Mac OS, you can find it at MacTex.
If you use Windows or Linux, go to
Latex-project.
To obtain such a TEX file with a header, click LATEX-sample .
Make a copy to play with, once you have downloaded TEX onto your computer.
At first, don't modify anything above "\begin{document}".