Algebraic topology II (Math 2420)

Announcements

• (4/1) HW5 is posted below. It is due Friday 4/11.
• (3/28) Final project proposals are due Friday 4/4. This is your homework for the week! Instructions here. I put a list of possible ideas to get you started. You are free to choose something else.

Course information

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

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 Kassar 105

Office hours:

• Tuesday 10:30-11:30
• Wednesday 4-5, or 5-6 in weeks with a department colloquium
• or by appointment

Important dates:

• Midterm 1: Week of March 3 (tentative)
• Midterm 2: Week of April 14 (tentative)
• Final projects: presented in class during reading period

Homework

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

Collaboration/academic honesty: 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 1/31). tex file, selected student solutions

HW2 (due 2/14). tex file, selected student solutions

HW3 (due 2/28). tex file, selected student solutions

HW4 (due 3/21). tex file

HW4 (due 4/11). tex file

Course materials

• For asynchronous discussions (e.g. questions about homework) we will use campuswire. Join here with access code 5653. This is the preferred way to ask questions about the homework or point out typos (rather than emailing me).

Final Project

Working in groups of 2, you’ll choose a topic and give an N-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 4)
• A project outline (due April 11)
• A draft of final presentation slides or talk notes (due April 18)

Some project ideas:

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

Topic schedule (subject to change)

• Week 1: Cohomology (Hatcher Section 3.1)
  • Thurs (1/23). Definition of cohomology, universal coefficient theorem
• Week 2: No class this week (I’m away at a conference)
  • Tues (1/28). Please read Hatcher Section 3.1.
  • Thurs (1/30).
  • Fri (1/31). HW1 due
• Week 3: Kunneth theorem (Hatcher 3.2)
  • Tues (2/4). Homology of a product, tensor products and Tor, algebraic Kunneth theorem
  • Thurs (2/6). Kunneth theorem, Eilenberg-Zilber theorem, acyclic models
  • Fri (2/7).
• Week 4: Cup products (Hatcher 3.2)
  • Tues (2/11). Cup products, examples
  • Thurs (2/13). Poincare duality
  • Fri (2/14). HW2 due
• Week 5: Poincare duality (Hatcher 3.3)
  • Tues (2/18). No class (university holiday)
  • Thurs (2/20). Orientations, Poincare duality
  • Fri (2/21).
• Week 6: Poincare duality (Hatcher 3.3),
  • Tues (2/25). Orientations, cup product and algebraic intersections
  • Thurs (2/27). Cup products and Poincare duality: computations and examples
  • Fri (2/28). HW3 due
• Week 7:
  • Tues (3/4). cohomology applications: spheres as H-spaces, manifolds that bound
  • Thurs (3/6). Take home-midterm (no class)
  • Fri (3/7). Midterm due by 5pm
• Week 8: Hatcher 4.1, 4.2
  • Tues (3/11). Homotopy classes of maps, homotopy groups
  • Thurs (3/13). Topology of function spaces, H-groups
  • Fri (3/15).
• Week 9: Hatcher 4.1, 4.2
  • Tues (3/18). relative homotopy groups, LES in homotopy, low-degree homotopy groups of spheres
  • Thurs (3/20). No class
  • Fri (3/22). HW4 due

• Week 10: Spring Break

• Week 11:
  • Tues (4/1). local PL lemma, LES in homotopy, fibrations
  • Thurs (4/3). fibrations, LES of fibration
  • Fri (4/4). final project proposal due
• Week 12:
  • Tues (4/8). homotopy excision, cellular approximation for maps
  • Thurs (4/10). Freudenthal suspension, Whitehead’s theorem
  • Fri (4/11). HW5 due, final project outline due
• Week 13: Midterm 2 this week
  • Tues (4/15). Whithead’s theorem, cellular approximation for spaces
  • Thurs (4/17). Hurewicz theorem
  • Fri (4/18). final project slides/notes due
• Week 14:
  • Tues (4/22).
  • Thurs (4/24).
  • Fri (4/25).
• Week 15: Final project presentations
  • Tues (4/29).
  • Thurs (5/1).

End of course.