Math 215a: Algebraic topology

• (12/12) Here are Ka Choi's notes on the lectures. Many thanks to him for taking these notes and letting me post them here.
• (8/29) I will be away at a conference next week. Professor Jones has kindly agreed to give the lecture on Wednesday 9/5. I think he will discuss applications of the fundamental group to knot theory. This should be a real treat. The lecture on 9/7 is cancelled; I will try to make it up later in the semester at a mutually convenient time.
• (8/24) Welcome to the Math 215a webpage! This course will be similar to what I taught two years ago. However we will do a bit more with smooth manifolds, and so now Bredon is a required text in addition to Hatcher. Also, unlike last time, homework will be graded by a GSI instead of by fellow students; since you will not have to spend time grading each other's homework assignments, I will give you a few more exercises to do.

Instructor: Michael Hutchings, 923 Evans, [my last name with the last letter deleted]@math.berkeley.edu. Tentative office hours (may be rescheduled some weeks): Wednesday 9-12. Outside of office hours, the best way to reach me is to send me an email. I generally check email once per day.

GSI: Qin Li. Office hours: Friday 3-5pm, room 935 Evans.

Lectures: MWF 1-2, room 3 Evans.

Prerequisites: The only formal requirements are some basic algebra, point-set topology, and "mathematical maturity". However, the more familiarity you have with algebra and topology, the easier this course will be. I think that all the point-set topology we will need (and a lot more) is reviewed in Bredon, Chapter I, Sections 1-13.

Syllabus: Algebraic topology seeks to capture key information about a topological space in terms of various algebraic and combinatorial objects. We will construct three such gadgets: the fundamental group, homology groups, and the cohomology ring. We will apply these to prove various classical results such as the classification of surfaces, the Brouwer fixed point theorem, the Jordan curve theorem, the Lefschetz fixed point theorem, and more.

An important topic related to algebraic topology is differential topology, i.e. the study of smooth manifolds. In fact, I don't think it really makes sense to study one without the other. So without making differential topology a prerequisite, I will emphasize the topology of manifolds, in order to provide more intuition and applications.

Textbooks:

• Allen Hatcher, Algebraic topology. This is a great book. You can download it for free from this page, which also has some additional material. However no single book can do everything that one might want, and this book does very little with smooth manifolds. So we will also be using:
• Glen Bredon, Topology and Geometry. This book does a lot more with smooth manifolds than Hatcher, and I also prefer its treatment of some of the topics that are covered in Hatcher. If you don't want to buy it, it is on reserve in the library.

• HW #1, due 9/21. Solutions
• HW #2, due 10/3. Solutions
• HW #3, due 10/17.
• HW #4, due 10/29..
• HW #5, due 11/9.
• HW #6, due 11/28.
• HW #7, due 12/6 or 12/10. (Please see the instructions for turning it in.)

Tentative course outline (There is a lot of material here, so we might not get to all of it, and I will refer to the textbooks for details of some of the proofs. My goal will be to explain the important ideas.)

• Introductory lecture: Questions which algebraic topology can answer (or try to).
  • Classification questions (manifolds, knots...)
  • Topological existence results (e.g. fixed point theorems)
  
  
  
• The fundamental group and covering spaces. (mainly from Hatcher, Chapter 1; also Bredon, Chapter III)
  
  
  • The fundamental group.
    • Definition and basic properties.
    • Basic examples: R^n, S^1 (by lifting to R), products, S^n for n>1 (by using smooth approximation to miss a point, or by cutting S^n into two simply connected pieces whose intersection is connected).
    • Applications of the fundamental group of the circle:
      • The Brouwer fixed point theorem and Borsuk-Ulam theorem in two dimensions.
      • The so-called fundamental theorem of algebra.
    • The Seifert-van Kampen theorem for the fundamental group of the union of two spaces.
    • The fundamental group of a CW complex with one 0-cell has a presentation with one generator for each 1-cell and one relation for each 2-cell.
    • The braid group. (Not in either of our textbooks.)
    • The trefoil knot is knotted and the Hopf link and Borromean rings are linked. (Not discussed at all in Hatcher, and not much in Bredon either.)
    
    
  • Covering spaces.
    • Definition and discussion of basic examples, including the relation between finite covers of S^1 and permutations, coverings of the torus, the covering of any genus g>1 surface by the hyperbolic plane, the canonical double covering of any manifold by an oriented manifold, and the non-example of branched covers of Riemann surfaces.
    • Lifting paths to a covering space; the lifting criterion for maps to a covering space.
    • The classification of covering spaces over a reasonable space.
    • An application to algebra: any subgroup of a free group is free.
    
    
  • Brief introduction to higher homotopy groups: definition, easy properties, introduction to pi_n(S^n) (which will be discussed more later), why homotopy groups are hard in general.
  
  
• Homology. (Hatcher, Chapter 2; Bredon, Chapter IV)
  
  
  • Introducing singular homology.
    • Warmup definition: simplicial homology of a Delta-complex.
    • Main definition: singular homology of a topological space.
    • H_0 is a direct sum of Z's, one for each path component.
    • Computation of the homology of a contractible space, using cones over simplices.
    • H_1 of a path connected space is the abelianization of pi_1.
    • Homology is a homotopy invariant.
  • Mayer-Vietoris sequence:
    • statement and simple examples, including the homology of S^n.
    • A bit of homological algebra: a short exact sequence of chain complexes induces a long exact sequence in homology.
    • Proof of Mayer-Vietoris sequence using subdivision lemma.
    • Understanding the connecting homomorphism in the Mayer-Vietoris sequence.
    • Application: proof of the generalized Jordan curve theorem.
    • An interesting example: linking numbers.
  • How to compute the homology of a CW complex.
    • Relative homology; long exact sequences of a pair and a triple; excision; isomorphism between relative homology and reduced homology of the quotient for "good pairs".
    • Equivalence of simplicial and singular homology.
    • Extensive discussion of the degree of a map from a sphere to itself.
    • Cellular homology of a CW complex.
    • Application to the "hairy ball theorem".
    • Isomorphism between singular and cellular homology of a CW complex.
    • Examples including real and complex projective space.
  • Euler characteristic, and Lefschetz fixed point theorem for simplicial complexes.
  • Homology with coefficients, Tor, and the universal coefficient theorem.
  • The Kuenneth formula for the homology of the product of two spaces, via the Eilenberg-Zilber theorem.
  
  
• Cohomology. (Hatcher, Chapter 3; Bredon, Chapters V and VI)
  
  
  • Definition of cohomology, and universal coefficient theorem for cohomology.
  • Definition of the cup product.
  • Statement of Poincare duality.
  • Computation of the cohomology ring of projective spaces.
    • Application: the Borsuk-Ulam theorem and the Ham Sandwich Theorem.
  • Important ingredients in the proof of Poincare duality, including the orientation bundle, compactly supported cohomology, direct limits, and the cap product.
  • Review/crash course on smooth manifolds.
  • If time permits, review/crash course on de Rham cohomology, and isomorphism of the latter with singular cohomology with real coefficients.
  • Statement of the geometric interpretation of cup product via intersection theory in a smooth manifold.
    • Application: the Lefschetz fixed point theorem on a smooth manifold (which actually counts fixed points of a generic smooth map with signs, rather than merely proving that a fixed point exists).
  • Poincare-Lefschetz duality for a manifold with boundary.
    • Application: the boundary of a 4n+1 dimensional oriented compact manifold has signature zero.
  • Alexander duality.
    • Application: the Klein bottle does not embed in R^3 (and more generally, a nonorientable closed n-manifold does not embed in R^(n+1)).