Algebraic toplogy II
Math 540b, USC Spring 2026

Instructor.

Kyler Siegel (kyler.siegel@usc.edu)

Time and location.

4:35pm - 5:50pm on Mondays and Wednesdays in 427 Kaprielian Hall (KAP).

Registration.

Please email Kyler if you are interested in the course and would like to receive email announcements but are not currently registered.

Office hours.

TBA (in the meantime feel free to reach out by email to schedule a meeting)

Prerequisites.

We will assume familiarity the material from Math 540a and Math 535a.

About.

This is a second semester graduate level course in algebraic topology. We will cover various advanced topics, sampling many of the highlights from the 20th century while building the foundations for modern research in topology.

Possible topics.

Poincaré duality
generalized cohomology theories
stable homotopy theory
vector bundles
principal bundles and their classifying spaces
Eilenberg-MacLane spaces
equivariant cohomology
characteristic classes
cohomology operations
topological K theory
Bott periodicity
spectral sequences
cobordism theory

Textbooks and references.

Algebraic Topology - Hatcher [H1]
Characteristic Classes - Milnor and Stasheff [MS] (see also here for modern typesetting)
Vector Bundles and K-Theory - Hatcher [H2]
K-Theory - Atiyah [A]
Cohomology Operations and Applications in Homotopy Theory - Mosher and Tangora [MT]
A User's Guide to Spectral Sequences - McCleary [Mc]
An Introduction to Homological Algebra - Weibel [W]
Spectral Sequences in Algebraic Topology - Hatcher [H3]
You Could Have Invented Spectral Sequences - Chow [C]

Problem sets.

TBA

Grading scheme.

TBA

Tentative schedule

Note: this schedule (along with the above information) is tentative and will be continuously updated to adapt to the pace of the course. Please check back regularly for updates and problem set assignments.

$\#$	Date	Material	References
1	Monday 1/12/26	Topological manifolds and Poincaré dualities. Geometric argument for surfaces using dual cell divisions. Local and global orientations. $R$-orientations. The orientable double cover of a topological manifold. Fundamental classes.	[H1] §3.3
2	Wednesday 1/14/26
	Monday 1/19/26: MLK Day - no class
3	Friday 1/23/26
4	Monday 1/26/26
5	Wednesday 1/28/26
6	Monday 2/2/26
7	Wednesday 2/4/26
8	Monday 2/9/26
9	Wednesday 2/11/26
	Monday 2/16/26: Presidents' Day - no class
10	Wednesday 2/18/26
11	Monday 2/23/26
12	Wednesday 2/25/26
13	Monday 3/2/26
14	Wednesday 3/4/26
15	Monday 3/9/26
16	Wednesday 3/11/26
	Monday 3/16/26: Spring Break - no class
	Wednesday 3/18/26: Spring Break - no class
17	Monday 3/23/26
18	Wednesday 3/25/26
19	Monday 3/30/26
20	Wednesday 4/1/26
21	Monday 4/6/26
22	Wednesday 4/8/26
23	Monday 4/13/26
24	Wednesday 4/15/26
25	Monday 4/20/26
26	Wednesday 4/22/26
27	Monday 4/27/26
28	Wednesday 4/29/26

Algebraic toplogy II Math 540b, USC Spring 2026