Math 215A

Algebraic Topology

Fall 2018

Tuesday, Thursday 10:30-11:50 am in 380-381T

Teaching Staff

Instructor Ralph Cohen Course Assistant Francois-Simon Fauteux-Chapleau
Office 380-383X Office 380-381-M
e-mail rlc@stanford.edu e-mail fsimon@stanford.edu
Office Hours Mondays and Wednesdays, 3:15 - 4:00 pm Office Hours Mondays and Wednesdays, 1:30 - 3:00 pm

Course Description

Math 215A will initiate the study of algebraic invariants of topological spaces up to homotopy equivalence . Through this quarter we will construct, prove properties about, and study applications of three such invariants:

  1. Fundamental Group. Covering spaces, the fundamental group of the circle, Van Kampen's theorem, the relation between covering spaces and the fundamental group.
  2. Homology. Simplicial, singular, and cellular homology. Equivalence of definitions. Excision and the Mayer-Vietoris sequence. Classical applications (Jordan curve separation, Borsuk-Ulam, Brouwer-fixed point, etc.).
  3. Cohomology. Cup product. The Universal Coefficient Theorem, and the Kunneth formula. Poincaré duality.

As a necessary ingredient, we will develop techniques in homological algebra. A basic class of topological spaces we will apply our discussion to is the class of CW complexes.

Textbook

Allen Hatcher's Algebraic Topology, available for free download here. Our course will primarily use Chapters 0, 1, 2, and 3.

Prerequisites

In addition to formal prerequisites, we will use a number of notions and concepts without much explanation. Topologically: you should be intimately familiar with point-set topology, in particular various constructions on spaces, the product and quotient topologies, continuity, compactness. Algebraically: groups, rings, homomorphisms, equivalence relations, quotient sets, quotients of groups by normal subgroups.

Course Grade

The course grade will be based on the following:

Homework Assignments

Homework will be posted here on an ongoing basis (roughly a week before they are due) and will be due at 5pm on the date listed, in the Course Assistant's mailbox. Late homeworks will not be accepted.

You are encouraged to discuss problems with each other, but you must work on your own when you write down solutions.

Due date Assignment
Thursday, October 4 Homework 1. Solutions.
Thursday, October 11 Homework 2. Solutions.
Thursday, October18 Homework 3. Solutions.
Thursday, October 25 Homework 4. Solutions.
Thursday, November 8 Homework 5. Solutions.
Thursday, November 15 Homework 6. Solutions.
Thursday, November 29 Homework 7. Solutions.

Lecture Plan

Date Lecture topics Book chapters Remarks
Sept. 25 Introduction: Spaces, maps, and homotopies. The fundamental group. Dependence on base point. Chapters 0 (first subsection), 1.1
Sept. 27 Covering spaces and lifting properties. The fundamental group of the circle. Functoriality. Chapters 1.1, 1.3 (first few pages)
Oct. 2 Fundamental groups of spheres. Van Kampen's theorem. Chapters 1.1, 1.2
Oct. 4 Fundamental groups of CW complexes. Chapter 0 (for definitions of CW complexes), 1.2
Oct. 9 Covering spaces I: Lifting properties, the universal cover, the correspondence theorem. Chapter 1.3
Oct. 11 Covering spaces II: The correspondence theorem, Deck transformations, examples. Graphs and sub-groups of free groups. Chapters, 1.3, 1.A
Oct. 16 The idea of homology, Delta-complexes, simplicial homology. Ch. 2 intro, 2.1 through p. 107.
Oct. 18 Singular homology, exact sequences and relative homology. The fundamental group and the first homology group. 2.1 pp 108-110, 113-119, section 2.A
Oct. 23 Homotopy invariance, excision - statement and applications, calculations Chapter 2.1, pp110-113, 119, 124-126
Oct. 25 Proof of excision via barycentric subdivision, the equivalence of singular and simplicial homology and the 5-lemma Chapter 2.1 pp119-124, 127-131, additional notes Additional notes completing the construction of the (Barycentric) subdivision proposition.
Oct. 30 The Mayer Vietoris sequence. Homology of good pairs. Degrees of maps between spheres. section 2.2
Nov. 1 More on the degree of a map. Classical applications (Brouwer's fixed point theorem, Jordan-Brouwer curve separation). Towards the homology of CW complexes. Chapter 2.2, 2.B
Nov. 6 Cellular (CW) homology groups. Immediate applications, including the homology of complex projective spaces. The cellular boundary formula, and applications to real projective space. Chapter 2.2
Nov. 8 The homology of real projective space. Homology with coefficients. Co-chain complexes and cohomology. Chapters 2.2, 3
Nov. 13 Singular cohomology groups and the Universal Coefficient Theorem. Chapter 3.1
Nov. 15 The Universal Coefficient Theorem continued. Cup product, and the cohomology ring. Statement of the cohomology ring of projective spaces, an application to the Borsuk-Ulam theorem. Chapters 3.1, 3.2
Nov. 27 Cup product in the relative setting. Calculation of the cup product on projective spaces, via a reduction to Euclidean spaces. Chapter 3.2
Nov. 29 The Künneth Theorem. An introduction to manifolds and orientations. Chapters 3.2, 3.3
Dec. 4 Manifolds and orientations in homology. The fundamental class. Chapter 3.3
Dec. 6 Orientations in homology continued. Poincaré duality and its applications. Chapter 3.3

Midterm and Final Exams

Here is the midterm exam. It is due on Thursday, November 1, at 5:00 pm. Please submit your completed exam as a pdf file via email to both Prof. Cohen and Francois-Simon.

Here is the Final Exam . It is due on Thursday, Dec.6 at 5:00 pm. Please submit your completed exam as a pdf file via email to both Prof. Cohen and Francois-Simon.