Math 541

Math 541: Introduction to Algebraic Topology II

Spring 2025

Instructor: Kristen Hendricks

Email: kristen. hendricks at rutgers.edu

Office Hours: Monday 10-11 in Hill 515, or by appointment. For small questions, note that I can be found at department tea most days other than Thursdays.

A printable copy of the syllabus is here. The homework sheets and lecture notes for this class live here.

Location and Time: MTh 8:30-9:50, in Hill 525.

Prerequisites: Having read and understood Chapters 0-2 of Hatcher (or equivalent).

Topics: This is a second course in algebraic topology. We will cover cohomology, homotopy theory, and some vector bundle theory as time allows.

Notes: Lecture notes will typically be posted at the link above a few hours before lecture, and updated afterward to with any corrections or changes to the amount of material covered in class.

Assignments: Homework exercises will be distributed each Monday in class and due on Canvas the following Monday at midnight. There will be twelve homework assignments in total. You are encouraged to work in groups on your homework, but you must write up all solutions yourself and cite your collaborators. You are also welcome to use any online resource you like as long as it is cited. (You are further encouraged to remember that not all online discussion of Hatcher’s exercises is correct.)

Texts: Reading will be announced in advance of the lecture on the homework sheets. Texts will be as follows.

No late homework will be accepted, but some number of your homework scores will be dropped in computing your grade. To have one score dropped, you must attend at least eleven lectures of the course; to have two scores dropped, you must attend at least twenty-two lectures of the course.

Assignments: There will be a take-home midterm distributed on March 10 and due on March 24. There will not be a final exam.

Grades: Grades will be computed based on 50% homework and 50% midterm.

Texts:

Cohomology and Homotopy Theory: A. Hatcher, Algebraic Topology; J. Milnor, Topology from the Differentiable Viewpoint

Vector Bundles and Characteristic Classes: J. Milnor and W. Stasheff, Characteristic Classes; A. Hatcher, Vector Bundles and K-Theory

Motivation: Here are a few questions which can be answered or helpfully rephrased using the techniques of this course.

Suppose that two spaces possess the same homology groups, as for example S^2 v S^4 and CP^2 do. How can we distinguish between them using the tools of algebraic topology?
What is special about manifolds, as opposed to arbitrary topological spaces? What can we say about them using the basic techniques we have developed?
Given a closed manifold M, a framed submanifold N is an embedded submanifold together with a smoothly varying basis for the normal bundle at each point in the set. What can one say about the set of such submanifolds, up to framed cobordism in M? How does this change if the framing condition is removed?
Given a smooth manifold M of dimension m, what are the smallest numbers n and k such that M may be immersed into n-dimensional Euclidean space and embedded into k-dimensional Euclidean space?
For what values of n is there a bilinear multiplication on R^n without zero divisors? We are aware of n=1 (the real numbers), n=2 (the complex numbers), n=4 (the quaternions), and n=8 (the octonions). Are there any others?
How can one distinguish between smooth manifolds that are homeomorphic but not diffeomorphic?