MATH 480 - Homotopy Theory
- Course Meetings: MWF 12:30 - 1:20 in DEM 002
- Office Hours: Thursday 12-1 and Friday 2:30-3:30 in PDL C-114
- Lecture Notes
- Final Project instructions and topics
- Recommended texts:
- Assortments of A Concise Course in Algebraic Topology by Peter May
- Chapter 1 of Spectra and stable homotopy theory by Cary Malkiewich
- Chapters 1 and 4 of Algebraic Topology by Allen Hatcher
- Chapters 7, 8, and 11 of Introduction to Topological Manifolds by Jack Lee
- Assortments of Category Theory in Context by Emily Riehl
- Assortments of Topology: A Categorical Approach by Tai-Danae Bradley, Tyler Bryson, and John Terilla
Course Description
A fundamental question in topology is the following: given two topological spaces $X$ and $Y$, how can one determine if there is an equivalence $X \cong Y$? One way to probe this question is to study algebraic invariants of X and Y. For example, there is an invariant called the fundamental group of a space, denoted by $\pi_1(-)$, and if $\pi_1(X) \not\cong \pi_1(Y)$, then in fact $X \not\cong Y$.
However, rather than merely assigning algebraic invariants, we can also study nice properties of the spaces themselves. For example, if one can show that a continuous function $∗ → X$ is a cofibration, then in fact $*\cong X$. These two perspectives, of using algebraic invariants to study spaces and exhibiting properties of spaces which are invariant, combine together to create homotopy theory.
This course will be an introduction to the methods used in homotopy theory. We will study algebraic invariants of spaces called homotopy groups and learn basic tools for calculation. Along the way, we will develop ways to treat topological spaces themselves as a more algebraic gadget, with an eye towards stable homotopy theory. Towards this goal, we will introduce a language known as category theory that will allow us to pass between and directly compare topology and algebra. This course will end with a final presentation on a topic of the your choice.
Week 1
Monday, March 30
- Syllabus day and background/motivation
- Notes for Lecture 1
Wednesday, April 1
- Finishing background/motivation, a discussion of the homotopy groups of spheres
- Notes for Lecture 2
- HOMEWORK 1 HERE, due Wednesday, April 8 in class
Friday, April 3
- Paths, loops, based spaces, and the fundamental group
- Notes for Lecture 3
Week 2
Monday, April 6
- The fundamental group of a path-connected space, starting to prove that $\pi_1(S^1) \cong \mathbb{Z}$
- Notes for Lecture 4
Wednesday, April 8
- Proved that $\pi_1(S^1) \cong \mathbb{Z}$
- Homework 1 due in class
- Notes for Lecture 5
- HOMEWORK 2 HERE, due Wednesday, April 15 in class (Note: Problem 4 is now optional)
Friday, April 10
- Substitute lecturer: Alex Waugh
- $\pi_1(X \times Y) \cong \pi_1(X) \times \pi_1(Y)$, fundamental group respects homotopy equivalence
- Notes for Lecture 6
Week 3
Monday, April 13
- Substitute lecturer: Alex Waugh
- Brouwer fixed point theorem and Borsuk-Ulam Theorem
- Notes for Lecture 7
Wednesday, April 15
- Homework 2 due in class (Note: Problem 4 is now optional)
- Substitute lecturer: Jay Reiter
- Start of category theory, definition and examples of categories
- Notes for Lecture 8
- HOMEWORK 3 HERE, due Friday, April 22 in class
Friday, April 17
- Substitute lecturer: Jay Reiter
- Functors and representability
- Notes for Lecture 9
Week 4
Monday, April 20
- Review of categories and functors
- No new lecture notes
Wednesday, April 22
- Representable functors, natural transformations and the Yoneda Lemma
- Notes for Lecture 11
- HOMEWORK 4 HERE, due Friday, May 1
Friday, April 24
- The Yoneda embedding, cones and cocones, limits and colimits
- Notes for Lecture 12
- Homework 3 due in class
Week 5
Monday, April 27
- Examples of limits and colimits: products, coproducts, pushouts
- Notes for Lecture 13
Wednesday, April 29
- More limits and colimits, together! In class!
- Notes for Lecture 14
- HOMEWORK 5 HERE, due Friday, May 8
Friday, May 1
- Formulas for limits and colimits of sets, pullbacks
- Notes for Lecture 15
- Homework 4 due in class
Week 6
Monday, May 4
- CW complexes, definition and some examples
- Notes for Lecture 16
Wednesday, May 6
- Sequential colimits, more examples of CW complexes and why we should care about them
- Notes for Lecture 17
- HOMEWORK 6 HERE
Friday, May 8
- Cofibrations
- Notes for Lecture 18
- CHOOSE TOPIC FOR FINAL PROJECT BY TODAY!!
- Fill out this form for final project
- Homework 5 due in class
Week 7
Monday, May 11
- Cofibers and Puppe sequences
- Notes for Lecture 19
Wednesday, May 13
- Long exact sequence induced by a Puppe sequence
- Notes for Lecture 20
Friday, May 15
- NO CLASS!!!!!
- Instead, you should go see Sean speak at the Undergrad Research Symposium! The session is from 11:30-12:45 in MGH 251.
- Homework 6 due
Week 8
Monday, May 18
- Fibrations
- Notes for lecture 21
Wednesday, May 20
- Hopf fibrations, path mapping space, factoring a morphism into a fibration followed by a weak equivalence
- Notes for lecture 22
Friday, May 22
- More on path mapping space, homotopy fiber, loop space, fiber Puppe sequence, long exact sequence associated to a fibration
- Notes for lecture 23
- In-progress final project draft due
Week 9
Monday, May 25
- MEMORIAL DAY, NO CLASS
Wednesday, May 27
- Adjunctions and examples
- Notes for Lecture 24
Friday, May 29
- Long exact sequence in homotopy groups, extended computation with the Hopf fibration
- Notes for Lecture 25
Week 10
Monday, June 1
- More on the long exact sequence
Wednesday, June 3
- Sharing final projects!!
Friday, June 5
- Final Project Due!!
- The Adams spectral sequence