# Algebraic topology II (Math 2420)

Graduate course, *Brown University*, 2024

## Announcements

- (4/23) Office hours this week are at the usual time (Tu 4:30, Th 3)
- (3/31) Office hours for the remainder of the semester will take place in my office Kassar 304.

## Course information

This is part two of a graduate course on algebraic topology. The main topics will be homotopy groups, cohomology, and Poincare duality.

### Course objectives

Prove foundational results, do computations, understand theoretical trends, see applications

## Prerequisites

Math 2410

## Textbooks

- Hatcher,
*Algebraic topology* - Bredon,
*Topology and geometry*

## Course expenses

Potentially none. Both textbooks are available online (Bredon through the Brown library webpage).

## Grading

homework 30%, 1st midterm 30%, 2nd midterm 30%, final project 10%

## Contact information

- Instructor: Bena Tshishiku (bena_tshishiku at brown.edu)

## Course events

Lectures: Tu-Th 9-10:20am in 85 Waterman Room 015

Office hours: Tuesday 4:30-5:30 and Thursday 3-4 in 111 Thayer Room 114

Important dates:

- Midterm 1: Week of March 11
- Midterm 2: Week of April 22
- Final projects: presented in class during reading period

## Homework

There will be weekly assignments posted below. The homework is designed to increase your engagement with the material, with your peers, and with me.

Collaboration: Please collaborate, and ask for help if you are stuck. You are required to write your solutions alone and acknowledge the students you worked with. If you find yourself writing down things that you can’t explain, you should go back and think more about the problem.

LaTeX: Homework solutions must be typed in LaTeX.

Late homework policy: For your homework grade, I will drop the score from your lowest assignment. View this as a one-time “get out of jail free card” in the event that you oversleep, forget, have a midterm, etc. As a general rule, late homework will not be accepted. If you have a medical emergency, I will ask for a note from a doctor or a dean. If you have an emergency that affects your ability to complete the coursework, please notify me as early as possible.

### Homework assignments.

HW1 (due 2/2). tex file, solutions

HW2 (due 2/9). tex file, student solutions

HW3 (due 2/16). tex file, solutions

HW4 (due 2/23). tex file, solutions

HW5 (due 3/1). tex file, solutions

HW6 (due 3/8). tex file, solutions

HW7 (due 3/22). tex file

HW8 (due 4/5). tex file, final project proposal

HW9 (due 4/12). tex file, final project outline

HW10 (due 4/19). tex file, final project slides

## Course materials

- For asynchronous discussions (e.g. questions about homework) we will use campuswire. Join here with access code 1053.

## Final Project

Working in groups of 2, you’ll choose a topic and give an 30-minute presentation during reading period.

The topic should be something related to the course that interests you.

As part of completing the final project, I will ask you to submit:

- A project proposal (due April 5)
- A project outline (due April 12)
- A draft of final presentation slides (or talk notes) (due April 19)

Some project ideas:

- Hopf invariant (Ella and Sujung)
- Grassmannians and classifying spaces (Xuyan and Roberta)
- Lefschetz fixed point theorem (Ivan and Edwin)
- group cohomology and group extensions (Ian and Semir)
- Nerve theorem and application (Carlos)
- H-spaces and Hopf algebras
- Local coefficients
- Brown representability
- spectra and (co)homology theories
- Steenrod squares
- Pontryagin-Thom theorem and stable homotopy groups
- cobordism groups
- J-homomorphism and stable homotopy groups
- Bott periodicity
- Alexander duality
- plus construction and (algebraic) K-theory
- vector bundles and topological K-theory
- Extra topics in Hatcher or Bredon or found elsewhere
- Manifolds are cell complexes

## Topic schedule (subject to change)

- Week 1:
**Homotopy classes of maps and mapping spaces**- Thurs (1/25). Homotopy classes of maps, homotopy groups, H-groups

- Week 2:
- Tues (1/30). Mapping spaces, H-group theorem
- Thurs (2/1). More mapping spaces, proof of H-group theorem
- Fri (2/2). HW1 due

- Week 3:
**Relative homotopy groups, LES of a pair**- Tues (2/6). LES of pair, low-degree homotopy of spheres, fibrations
- Thurs (2/8). LES of a pair proof, Hopf fibration
- Fri (2/9). HW2 due

- Week 4:
**Fibrations and fiber bundles**- Tues (2/13). snow day
- Thurs (2/15). LES of fibration, Serre vs. Hurewicz vs. fiber bundle
- Fri (2/16). HW3 due

- Week 5:
**Homotopy excision, Freudenthal suspension**- Tues (2/20). No class (university holiday)
- Thurs (2/22). homotopy excision, cellular approximation for maps
- Fri (2/23). HW4 due

- Week 6:
- Tues (2/27). Freudenthal suspension, Whitehead’s theorem
- Thurs (2/29). Whithead’s theorem, cellular approximation for spaces
- Fri (3/1). HW5 due

- Week 7:
**Whitehead and Hurewicz theorems**- Tues (3/5). Whitehead theorem proof, Hurewicz theorem
- Thurs (3/7). Hurewicz theorem proof
- Fri (3/8). HW6 due

- Week 8:
**Cohomology (and midterms)**- Tues (3/12). Cohomology introduction
- Thurs (3/14). Hom and Ext
- Fri (3/15). (no homework due this week)

- Week 9:
**Universal coefficients**- Tues (3/19). Universal coefficient theorem discussion, Ext computation
- Thurs (3/21). Universal coefficient proof, application of Ext to group cohomology
- Fri (3/22). HW7 due

Week 10: Spring Break

- Week 11:
**Kunneth theorem**- Tues (4/2). Homology of product, Tor, algebraic and geometric Kunneth theorems
- Thurs (4/4). Eilenberg-Zilber theorem, acyclic models method
- Fri (4/5). HW8 due, final project proposal due

- Week 12:
**Cup products**- Tues (4/9). Construction of the cup product, examples
- Thurs (4/11). Cup product properties and examples
- Fri (4/12). HW9 due, final project outline due

- Week 13:
**Poincare duality**- Tues (4/16). Poincare duality intro, orientations on manifolds
- Thurs (4/18). Fundamental class, PD and cup products
- Fri (4/19). HW 10 due, final project slides due

- Week 14:
**Midterm 2 this week**- Tues (4/23). PD and cap products, manifolds that bound
- Thurs (4/25). Brown representability
- Fri (4/26).

- Week 15: Final projects
- Tues (4/30). Edwin+Ivan, Sujung+Ella
- Thurs (5/2). Carlos, Roberta+Xuyan

- Week 16: Final projects
- Tues (5/7). Ian+Semir