Math 215A, Fall 2018

Instructor: James Conway, 933 Evans, conway [at]

Office Hours: Thursday 11:30AM - 1PM, and by appointment. I answer e-mails with relative frequency.

Lectures: Tuesday/Thursday 9:30AM (=9:40AM) - 11AM in 70 Evans

Syllabus: Algebraic topology seeks to capture as topological information about a space in terms of algebraic and combinatorial data. We will see three main constructions: the fundamental group, the homology groups, and the cohomology ring. We will use these to prove various classical results such as classification of surfaces, fixed point theorems, understand covering spaces, and more.

Textbook: Allen Hatcher's Algebraic Topology is freely available here. This will be our main resource, and we will cover much of Chapters 0-3. Additional material to be noted in the course material below.

Homework/Grading: You will have homework once a week, with several questions graded out of 2. You will have a takehome midterm and a takehome final (exact dates below). Your final grade will be 25% homework, 25% midterm, and 50% final.

What we actually did + homework

(red means extra important; underline means homework)

August 23: introduction to the course Reading: Hatcher, Chapter 0 August 28: more on CW complexes Reading: Hatcher, Chapter 0 Homework 1 (due September 4): Click here August 30: homotopy extension property; fundamental group is a group Reading: Hatcher, Chapter 1.1 September 4: fundamental group of the circle; lifting properties; applications Reading: Hatcher, Chapter 1.1 Homework 2 (due September 13): Click here September 6: induced maps; started covering spaces (see 'Reading' below) Reading: Hatcher, Chapter 1.1, 1.3 (including statement of 1.30, but not [yet] its proof) September 11: no class today — we will make up this class during RRR week September 13: lifting criterion and uniqueness of lifts; started construction of universal cover Reading: Hatcher, Chapter 1.3 Homework 3 (due September 20): Click here September 18: constructions and classification of based covering spaces for "nice" spaces Reading: Hatcher, Chapter 1.3 September 20: deck transformations; group actions; K(G,1) spaces Reading: Hatcher, Chapter 1.3, 1.B September 25: (substitute) Seifert–van Kampen theorem Reading: Hatcher, Chapter 1.2 September 27: more K(G,1) spaces Reading: Hatcher, Chapter 1.B Homework 4 (due October 4): Click here October 2: no class today — we will make up this class during RRR week October 4: introduction to homology; singular chain complexes Reading: Hatcher, Chapter 2.1 Homework 5 (due October 11): Click here October 9: first computations; chain maps and induced maps Reading: Hatcher, Chapter 2.1 October 11: homotopy invariance; exact sequences and snake lemma; subdivision lemma (no proof); Mayer–Vietoris Reading: Hatcher, Chapters 2.1 (pp 110-113, 119) and 2.2 (from p 149) Video: part of the proof of the Snake Lemma from the movie It's My Turn October 16: Mayer–Vietoris sequence; reduced homology Reading: Hatcher, Chapter 2.1 (p 110) and Chapter 2.2 (from p 149) October 16–18: take-home midterm October 18: excision; collapsing subsets Reading: Hatcher, Chapter 2.1 (from p 119) October 23: proofs of subdivision, excision, collapsing subsets Reading: Hatcher, Chapter 2.1 (from p 119) Homework 6 (due November 6): Click here October 25: degrees of maps between spheres; cellular homology, definition and calculations Reading: Hatcher, Chapter 2.2 October 30: more cellular homology calculations; cellular homology = singular homology proof Reading: Hatcher, Chapter 2.2 November 1: end of cellular homology = singular homology proof; Lefschetz fixed point theorem; Euler characteristic Reading: Hatcher, Chapter 2.2 (p 146) and Chapter 2.C November 6: homology with coefficients; tensor products, universal coefficient theorem, Tor Reading: Hatcher, Chapter 2.2 (from p 153) and Chapter 3.2 (p 218 on tensor products) and Chapter 3.A Homework 7 (due November 15): Click here November 8: more universal coefficient theorem; introduction to cohomology Reading: Hatcher, Chapter 3.1 November 13: calculating cohomology from homology; the Künneth formula Reading: Hatcher, Chapter 3.1 and Chapter 3.B November 15: proof of the Künneth formula for CW complexes Reading: Hatcher, Chapter 3.B Reading: Bredon, Chapter VI (Products and Duality) November 20: no class today — smoke day November 27: cup product Reading: Hatcher, Chapter 3.2 and Chapter 3.B (from p 278) Reading: Bredon, Chapter VI (Products and Duality) November 29: cup product; manifolds and orientations Reading: Hatcher, Chapter 3.2 and 3.3 December 4: Poincaré duality Reading: Hatcher, Chapter 3.3 Homework 8 (not due): Click here December 6: more Poincaré duality Reading: Hatcher, Chapter 3.3 December 6–12: take-home final (available on bCourses)