# Math 215A, Fall 2018

Instructor: James Conway, 933 Evans, conway [at] berkeley.edu.

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 to be determined). 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:
November 15:
November 20:
November 27:
November 29:
December 4:
December 6:
```

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