Spring 2023
Math 5329/7309: Topology I



Course format

This course will be primarily instructed face-to-face. On rare occasions we may switch to an online format.

Course description

This class is a graduate-level topology class, building on (the prerequisite) undergraduate topology. The main motif will be computations of homology.

Beyond a fluency with the above topics, another goal of this class is for you to become familiar with mathematical thinking---questioning and understanding why definitions exist, identifying when you or another communicator is being precise or imprecise (and for what purpose), developing tastes that are rooted in practice and informed experience, exploring the mathematical landscape on your own.

Prerequisite: MATH 4330 with a grade of C or higher.


Textbook and resources.

You do not need to buy a textbook for this class. The following is a freely available resource:

  1. Allen Hatcher's Algebraic Topology textbook.

Resources for undergraduate point-set topology (all freely available):

  1. Here is a link to the Fall 2023 version of Math 4330.
  2. Allen Hatcher's notes on point-set topology.
  3. Stefan Waner's notes on Elementary Topology.
  4. Sidney A. Morris's Topology without Tears.

Resources for proof (all freely available):

  1. Book of Proof by Richard Hammack. You will need to be comfortable with all the material in this book.
  2. Introduction to Proof in Analysis by Steve Halperin. You will need to be comfortable with Chapters 1-3 of this book.
  3. Introduction to mathematical arguments by Michael Hutchings. You will need to be comfortable with all this material.

The syllabus

The syllabus can be found here.

Important dates

Final Exam: Wednesday, May 1.

Collaboration policy

I strongly encourage all of you to collaborate. Please do so. If you do, you must indicate clearly on every assignment that you have collaborated, and indicate with whom. However, write solutions on your own. It is fine to think through problems and find solutions with each other, but when it comes to the act of writing your homework, you must do so without assistance from another. This is because the act of solving something and writing a mathematical proof are two different skills, and I want you to also hone the latter. As an extreme anti-example, copying and pasting solutions/proofs will not be tolerated. To reiterate, you may not write solutions together.



Readings (read before class)

  1. Wed, Jan 17. A rapid review of point-set topology.
  2. Mon, Jan 22. (Hiro not in class, Dr. Hirsh substituting) Exercises on open subsets of Euclidean space.
  3. Wed, Jan 24. (Hiro not in class, Dr. Hirsh substituting) Exercises on open covers.
  4. Mon, Jan 29. Review of abelian groups, homomorphisms, isomorphisms, kernels, images.
  5. Wed, Jan 31. Homology's basic properties. Updated slightly as of Feb 1st. Notes from a classmate.
  6. Mon, Feb 5. Homotopy equivalence. Updated slightly as of Feb 1st. Notes from a classmate.
  7. Wed, Feb 7. Mayer-Vietoris Notes from a classmate.
  8. Mon, Feb 12. Mayer-Vietoris practice: Some graphs. Notes from a classmate.
  9. Wed, Feb 14. Mayer-Vietoris practice: Pants. Notes from a classmate.
  10. Mon, Feb 19. Review day. Notes from a classmate.
  11. Wed, Feb 21. Application: Invariance of domain and Brouwer Fixed Point Theorem.Notes from a classmate.
  12. Mon, Feb 26. Review exercises on quotient spaces. Hiro not in class. Dr. Hirsh will lead the class.
  13. Wed, Feb 28. More review exercises on quotient spaces. Hiro not in class. Dr. Hirsh will lead the class.
  14. Mon, Mar 4. CW complexes. Notes from a classmate.
  15. Wed, Mar 6. Real projective space. Notes from a classmate.
  16. Mon, Mar 18. RPn as a CW complex.Notes from a classmate.
  17. Wed, Mar 20. Toward cellular homology. Notes from a classmate. Notes from another classmate.
  18. Mon, Mar 25. Toward cellular homology some more. Notes from a classmate.
  19. Wed, Mar 27. The higher differentials in the cellular chain complex. Notes from a classmate.
  20. Mon, Apr 1. Degree d maps between spheres. Notes from a classmate.
  21. Wed, Apr 3. Homology of real projective space Notes from a classmate.
  22. Mon, Apr 8. Class will be asynchronous. We will neither meet physically nor simultaneously. A video will be posted, and this material will not appear on the Final Exam. Proof that cellular homology computes homology.

    The lecture can be found here. You must log in with your TXST net id to access it. There is a bit of text lag; please be patient. I will often talk about an object, and the object will only show up seconds later due to lag in the Zoom Whiteboard feature.
  23. Wed, Apr 10. Computations of homology.Notes from a classmate.
  24. Mon, Apr 15. Computations of homology, continued.Notes from a classmate.
  25. Wed, Apr 17. Singular homology. Notes from a classmate.
  26. Mon, Apr 22. Euler characteristic. Notes from a classmate. More notes from a classmate.
  27. Wed, Apr 24. More on singular homology.Notes from a classmate.
  28. Mon, Apr 29. Last day of class. What's next?


Homeworks

All assignments are due on Canvas, and must be uploaded in PDF format.
All homework assignments may be shared with your classmates, so I recommend that you remove your names from your scanned/typed/written assignments. (When you upload your homework, I will know which assignments belong to whom, thanks to Canvas.)