Graduate Topology, FSU, Spring 2025

Class: TTh 9:45-11 AM, Classroom: LOV 105

Syllabus: Algebric topology, with special focus on simplicial and singular homology and cohomology.

Office hours: Tuesdays 11-12, or by appointment. Office: LOV 313.

Textbook and Resources

Algebraic Topology by Allen Hatcher

We may supplement from other textbooks and notes down the line, and this webpage will be updated to reflect that.

Important dates: Final exam: TBD

Graduate Topology, FSU, Fall 2024

Class: MWF 10:40-11:30 AM, Classroom: LOV 105

Syllabus: Quick brush-up of basic point-set topology, followed by fundamental groups and covering spaces.

Office hours: Fridays 9:40-10:40, or by appointment. Office: LOV 313.

Important dates: October 16-18 (Midterm); December 11-13 (Final)

Textbook and Resources

Textbooks and other supplementary materials are all legally available for free.

Core materials:

• Topology by James Munkres
• Algebraic Topology by Allen Hatcher
• An Introdocution to Algebraic Topology by Joseph Rotman
• Algebraic Topology: A Comprehensive Introduction by Laurentiu Maxim

Note that despite using the word 'textbook' we are not going to follow them word for word, or cover to cover.

We will follow Munkres for our basic topology recap: which will primarily involve covering Chapters 2 and 3, and if time permits then parts of Chapters 4 and 7.

For fundamental groups and covering spaces, we will learn from a mixture of the latter three sources listed above. They all have their own strengths. I will cherry-pick from Rotman, Hatcher and Maxim's notes for best results. Homework problems will be usually assigned from Hatcher, and sometimes from Rotman. I will try to upload notes on fundamental groups and covering spaces at least once a month: these notes will be, if not isomorphic, then at least (weakly?) homotopy equivalent to my lectures.

Other resources

• Topology without Tears by Sidney Morris
• Notes on Introductory Point-set Topology by Allen Hatcher
• An outline summary of basic point set topology by Peter May
• Notes on categories, the subspace topology and the product topology by John Terilla
• Parts VI and VII of Algebraic Topology: A First Course by William Fulton

Prerequisites for this course

For prerequisites, the webpage for the graduate program in pure math at FSU states: "Entering graduate students are expected to have taken at least two semesters of undergraduate courses in each of the subjects abstract algebra and real analysis. Semester long courses in general topology and complex analysis are also strongly suggested." Based on that I will assume complete familiarity with the following.

• Chapter 1 of Topology by James Munkres
• Book of Proofs by Richard Hammack
• Introduction to Proof in Analysis by Steve Halperin
• Introduction to mathematical arguments by Michael Hutchings
• The theory parts (only) of Chapters 1-5, 9-12, 14, 16-17 and 20-22 of Abstract Algebra: Theory and Applications by Thomas W. Judson

Some familiarity with Chapters 2-3 of Munkres's Topology will be helpful, but not necessary.

Collaboration policy

Collaboration with your peers on the homeworks is not only allowed, but in fact strongly encouraged. However, please write solutions on your own. Please also indicate very clearly the names of all you collaborated with on that homework.

One of the aims of a grad course is for students to learn how to write mathematics: which is a very different skill from thinking mathematics. So think together, but write on your own. In particular, the extreme case of copying and pasting someone else's homework solutions will not be tolerated.

Collaborations are not allowed on the exams.