Math 215A, Algebraic Topology

UC Berkeley, Fall 2017

Announcements:

Office hours on 10/26 will be from 9am-noon. Office hours for 11/2 will be rescheduled to 10/31 from 9am-noon.
My office hours on 9/14 will be from 10:30-12:30, instead of the usual 1:00-3:30.

Instructor:

Michael Hutchings
hutching@math.berkeley.edu
Office phone: 510-642-4329.
Office: 923 Evans.
Tentative office hours: Thursday 1:00-3:30.

Piazza:

In addition to my office hours, I have set up a piazza page for the course here. Please feel free to ask questions about the course content there.

Textbook:

The textbook for this course is Algebraic Topology by Allen Hatcher. The book with the latest corrections can be legally downloaded for free here.

Homework:

The course grade will be based on homework, which will usually be due a little less than once per week. Collaboration on homework is encouraged but must be acknowledged, and you must write your own solutions. Please write clearly, because your assignments will be peer-graded!

HW#1, due Sep. 6 in class: Hatcher section 1.1, problems 3, 5, 6, 7, 16, 20. Also, show that if G is a topological group (a topological space with a group structure such that inversion and multiplication are continuous), then pi_1(G,1) is abelian.
HW#2, due Sep. 15: Hatcher section 1.2, problems 3, 7, 10, 14, 17, 19; Hatcher section 1.A, problem 3.
HW#3, due Sep 25.
HW#4, due Oct 6.
HW#5, due Oct. 18: Hatcher section 2.1, problem 17; section 2.2, problems 2, 3, 7, 8, 11, 17.
HW#6, due Nov 1.
HW#7, due Nov 15.

Instructions for peer-grading homework: Write your name, the name of the student whose homework you are grading, and your feedback, all on a separate sheet. Each problem should get 3 points if it is more or less all correct, 2 points if it has some minor problems, 1 point if it has more serious problems, and 0 points if there is no significant progress toward a solution. You are welcome to ask about how to solve the prolems on piazza. Be sure to give constructive comments. An important goal of this exercise is to help everyone improve their mathematical communication skills. Graded homework should be returned one week after it is handed in.

Syllabus:

The goal of this course is to explain how algebraic topology works and how it can be applied to geometric problems. Specifically, we will study the basics of the fundamental group, homology, and cohomology, roughly corresponding to the first three chapters of Hatcher. I will skip some of the more specialized topics in Hatcher (especially in some of the appendices), and will introduce some more geometric topics which are not covered by Hatcher.

What we actually did in class:

Below I will list what we did in each lecture, and where you can read more about it.

(Wed 8/23) Introduction. Informal discussion of the classification of surfaces. (To prove all this rigorously takes a lot of work. I think it is done for example in the book Topology by Munkres.) We will start on the fundamental group next time.
(Fri 8/25) Definition of the fundamental group. Most of the computation of the fundamental group of the circle. See Hatcher section 1.1. (I just noticed that I used the letters s and t oppositely from Hatcher. Sorry for any confusion.) We will finish this computation and explain more about the fundamental group next time.
(Mon 8/28) Finished the calculation of the fundamental group of the circle. Proved the Brouwer fixed point theorem in 2 dimensions. Discussed the dependence of the fundamental group on the choice of base point. This is all still in Hatcher section 1.1.
(Wed 8/30) Homotopy invariance of the fundamental group. Statement of the Seifert-van Kampen theorem. The latter is explained in Hatcher section 1.2, and we will discuss it more next time.
(Fri 9/1) Examples of the Seifert-van Kampen theorem, and most of the proof.
(Wed 9/6) How to compute the fundamental group of a knot complement. CW complexes and their fundamental groups. See section 1.2. For more about CW complexes, see chapter 0 and the appendix at the end of the book.
(Fri 9/8) Introduced covering spaces. Proved the path lifting lemma and homotopy lifting lemma. Used these to show that a covering space induces an injection on the fundamental group. See section 1.3 of Hatcher.
(Mon 9/11) Started explaining the classification of covering spaces over a "reasonable" space.
(Wed 9/13) Proved the criterion for lifting maps to a covering space, and continued discussing the classification of covering spaces. (Almost finished.)
(Fri 9/15) Concluded the classification of covering spaces. Briefly introduced higher homotopy groups. Will start discussing homology next time.
(Mon 9/18) Simplicial homology of a delta complex. See Hatcher section 2.1.
(Wed 9/20) Definition of singular homology. Most of the proof that H_1 of a path connected space is the abelianization of pi_1. See Hatcher sections 2.1 and 2.A.
(Fri 9/22) Homology of a contractible space. Algebraic generalities about chain complexes.
(Mon 9/25) Homotopy invariance of singular homology. This is also in Hatcher section 2.1, but unlike Hatcher, I defined the "prism operator" using the technique of "acyclic models", in order to avoid getting distracted by irrelevant geometry of simplices. This technique will be very useful for some more complicated constructions to come later.
(Wed 9/27) Statement of the Mayer-Vietoris sequence, and computation of examples. See Hatcher section 2.2.
(Fri 9/29) Construction of the MV sequence.
(Mon 10/2) Used the MV sequence to prove the generalized Jordan Curve Theorem. (This is in section 2.B of Hatcher.) As a fun example of this, discussed linking number of knots in R^3. (I think this is not in Hatcher.)
(Wed 10/4) Relative homology. The relative homology of a "good" pair (X,A) is the reduced homology of X/A. See Hatcher section 2.1.
(Fri 10/6) Proof that the simplicial homology of a Delta complex agrees with its singular homology. See Hatcher section 2.1.
(Mon 10/9) Cellular homology of a CW complex. Degree of maps from S^n to S^n. See Hatcher section 2.2.
(Wed 10/11) More about degree of maps and cellular homology.
(Fri 10/13) The Euler characteristic and the Lefschetz fixed point theorem. See Hatcher section 2.C for technical details of simplicial approximation.
(Mon 10/16) More about the Lefschetz fixed point theorem. Started discussing homology with coefficients; see section 2.2 of Hatcher.
(Wed 10/18) The universal coefficient theorem for homology. This is in section 3.A of Hatcher.
(Fri 10/20) As an application of homology with coefficients, we proved the Borsuk-Ulam theorem and applied it to the ham sandwich theorem. We then started discussing the Kunneth formula.
(Mon 10/23) Proved the Kunneth formula for homology (skipping some routine steps). Unlike Hatcher, I followed the approach of the Eilenberg-Zilberg theorem and acyclic models. For details, see Bredon, Topology and Geometry, sections 4.16 and 6.1.
(Wed 10/25) Started discussing cohomology and cup product. See Chapter 3 of Hatcher.
(Fri 10/27) More about cup product.
(Mon 10/30) The cohomology cross product, and the cohomology ring of a Cartesian product.
(Wed 11/1) The fundamental class, and the statement of Poincare duality. See Hatcher section 3.3.
(Fri 11/3) More about Poincare duality.
(Mon 11/6) Compactly supported cohomology, and outline of the proof of Poincare duality.
(Wed 11/8) Easy corollaries of Poincare duality: nondegeneracy of the cup product pairing and examples. The Euler characteristic of a compact odd-dimensional manifold is zero.
(Mon 11/13) Review of smooth manifolds, in preparation for discussing the relation between cup product and intersection of submanifolds. (We have now left the world of Hatcher! The material reviewed today is in many textbooks, such as Introduction to Smooth Manifolds by John M. Lee.)
(Wed 11/15) Statement of the relation between cup product and intersection, and simple examples. See these notes.
(Mon 11/20) The Lefschetz fixed point theorem on a smooth manifold.
(Mon 11/27) More about the Lefschetz FPT and the Poincare-Hopf index theorem. Started to prepare to discuss de Rham cohomology. For more about the latter, I strongly recommend Differential forms in algebraic topology by Bott and Tu.
(Wed 11/29) Very brief review of differential forms. Introduction to de Rham cohomology and proof of its homotopy invariance.
(Fri 12/1) Proof of the de Rham theorem.