Fall 2013

Mathematics 231a

Algebraic Topology

Catalog Number: 7275
Instructor: Hiro Lee Tanaka
Half course (fall term). M., W., F., at 2.
EXAM GROUP: 7
Office: Science Center 341 (in back of the math library)
Office Hours: Wed 1-2 PM, Th 2-3 PM.
Phil's Office Hours: Tu, Th 5-6 PM.
Covering spaces and fibrations. Simplicial and CW complexes, Homology and cohomology, universal coefficients and Kunneth formulas. Hurewicz theorem. Manifolds and Poincare duality.
Prerequisite: Mathematics 131 and 132.

Course Announcements

  1. Take a moment to fill out the mid-semester survey. It's important for me to get feedback about the class so I know how we're doing. You'll also get a say in what you'll learn the week before Thanksgiving! Fill this out by Monday, November 4th.
  2. The informal reading group on oo-categories will meet Wednesdays, 3 PM to 4 PM (right after class), in Room 232 Science Center. See the web site for the reading group for more information.

Recommended resources

Two textbooks I'd recommend are:

  1. Hatcher, "Algebraic Topology". A lot of pictures, plenty of examples.
  2. May, "A Concise Course in Algebraic Topology". No pictures, few examples, but very clear explanation of the modern, structural aspects of algebraic topology.

Homeworks

Grading policy: As I said, you'll be graded on two components: (a) Comprehension, and (b) Effort. Every week Phil will tell you via homework comments where you seem to be having troubles. You must come see me or Phil to talk about the problems you didn't seem to understand--this is the effort component of your grade. When you've come to talk to us about every problem you had, you can consider that a perfect score on the effort spectrum of your grade.

Collaboration policy: The collaboration policy was outlined in the syllabus. I'll copy-paste it here: I strongly encourage all of you to collaborate. Please do so with exercises and homeworks and talk preparations. However, indicate clearly on every assignment that you have collaborated, and indicate with whom. (Professors or students or friends or parents, or whoever helped you in finding solutions.) Finally, write solutions on your own. It is fine to think through problems and find solutions with each other, but when it comes to the act of writing it all up, you must do so without assistance from another. As an extreme anti-example, copying and pasting solutions/proofs will not be tolerated.

  1. Homework Twelve Due Monday, Dec 2. Hand in every problem except problem 7(a).
  2. Homework Eleven Due Friday, Nov 22. Hand in every problem except problem 5.
  3. Homework Ten Due Friday, Nov 15th in class. Hand in every problem except problems 6(a), 7(c), and 7(d).
  4. Homework Nine Due Friday, Nov 8th in class. Hand in every problem.
  5. Homework Eight Due Friday, Nov 1st in class. Hand in every problem.
  6. Homework Seven Due Friday, Oct 25th in class. Hand in every problem.
  7. Homework Six Due Friday, Oct 18th in class. Hand in every problem.
    • A correction to Homework 6: I fixed the range in the Freudenthal Suspension Theorem statement.
  8. Homework Five Due Friday, Oct 11th in class. Hand in every problem.
    • A correction to Homework 5: In Problem 6, you may assume Q is just a closed line interval.
    • A correction to Homework 5: In Problem 5(a), I've changed the problem to just proving the existence of a deformation retract.
    • A correction to Homework 5: In problem 3, the infinitely iterated retraction wasn't fully defined because r_k isn't defined for points on A in higher cells. I think I fixed that. Otherwise, you can hopefully figure out what I mean.Please download the new PDF. I've made a change to problem 3 and 3(g), and all the corrections to homework 5 so far have been highlighted in blue color for easy detection.
    • A correction to Homework 5: In the definition of HEP, we should also require that F(x,0) = f(x) for every x. The new PDF reflects all corrections.
    • A correction to Homework 5: In Problem 1(a) and 1(b), you only need to show that X x [0,1] *retracts* onto the space A x [0,1] \cup X x {0}. "Deformation" retract is too strong a condition. (As a matter of definition, we say that a subspace Y of a space X is a "retract" of X if there exists a continuous map from X to Y such that the composition Y --> X --> Y is the identity map. This is a weaker condition than being a deformation retract of X.)
  9. Homework Four Due Friday, Oct 4th in class. Hand in every problem.
    • A correction to Homework 4: In problem 2, assume that there exist neighborhoods of x_0 and y_0 that deformation retract to x_0 and y_0. A stronger (but often satisfied) condition is that X and Y are locally contractible--that is, *any* point has an open neighborhood that can be contracted to the point.
    • A correction to Homework 4: Obviously, K_0 has to be a free Abelian group (since H_0 of a space is always free abelian). However, do *not* assume that any of the K_n are finitely generated. You can have crazy, infinitely-generated abelian groups. (As a hint: Of course, if you have an abelian group generated by some giant set Q, you could just try wedging together a collection of spaces indeed by Q, for instance.)
  10. Homework Three Due Friday, Sept 27th in class. Hand in every problem.
    • A correction to homework 3: In problem 8(a)(ii), F(v,0) should equal id(v), not some undefined function f.
    • As a reminder, C_n(X) means the nth part of the singular chain complex of X. So it's the free abelian group generated by maps from the n-simplex to X.
  11. Homework Two Due Friday, Sept 20th in class. Hand in: 1, 2, 3, 6, 8, 9, 10, 12, 15, 16.
  12. Homework One Due Friday, Sept 13th in class. The problems you must hand in are: 3, 5, 8, 9, 11, 13, and 14.

Class Notes

These are notes I prepare ahead of class. Due to time constraints and questions, the material I cover in class is often only part of what I prepare; at the same time, the questions in class can bring out new topics I haven't written in these notes. But everything necessary for homework is contained in these notes.

  • Lecture 36. Poincare Duality. Monday, December 2.
  • Lecture 35. Van Kampen Theorem, guest lecture by Omar. Monday, November 25.
  • Lecture 34. Lefschetz Fixed Point Theorem, guest lecture by Gijs. Friday, November 22.
  • Lecture 33. Cohomology as maps into K(G,n), guest lecture by Phil. Wednesday, November 20.
  • Lecture 32. Cohomology ring of CP^oo, guest lecture by Phil. Monday, November 18.
  • Lecture 31. Eilenberg-Steenrod Axioms for cohomology. Relative cup product and cross product. Cohomology with coefficients if time allows. Friday, November 15.
  • Lecture 30. Relative cohomology and basic properties. Wednesday, November 13.
  • Lecture 29. Cup product. Monday, November 11.
  • Lecture 28. Universal coefficient theorem. Ring structure on cohomology. Friday, November 8.
  • Lecture 27. Kunneth formula continued, and universal coefficient theorem. Cohomology. Wednesday, November 6.
  • Lecture 26. Kunneth formula, homology with coefficients, and product CW structures. Monday, November 4.
  • Lecture 25. Friday, November 1.
  • Lectures 23 and 24. Products and coproducts. Hurewicz Theorem. Monday, October 28 and Wednesday, October 30.
  • Lecture 22. H_n and pi_n commute with sequential colimits. Friday, October 25th.
  • Lecture 21. Limits and colimits. Wednesday, October 23rd.
  • Lecture 20. Topology of mapping spaces. Monday, October 21st.
  • Lecture 19. Introduction to compactly generated spaces. Friday, October 18th.
  • Lectures 17 and 18. Fiber bundles. Friday, October 11th and Wednesday, October 16th.
  • Lecture 16. Brief discussion of homotopy excision. LES of a fibration. Wednesday, October 9th.
  • Lecture 15. Mapping Cylinders, Whitehead's Theorem. Monday, October 7th.
  • Lecture 14. Compression Lemma. Proof that any weak homotopy equivalence is an isomorphism on homology. Friday, Oct 4th.
  • Lecture 13. Finish LES of relative homotopy groups, base point change, homotopy v homology. Wednesday, Oct 2st.
  • Lecture 12. pi_n as a functor, relative homotopy groups, long exact sequence of homotopy groups. Monday, Sept 30.
  • Lecture 11. Homotopy groups. Friday, September 27th.
  • Lecture 10. Cellular homology. Wednesday, September 25th.
  • Lectures 8, 9. Reduced homology, CW complexes. RP^n and CP^n.
  • Lecture 7. Long exact sequences in homology, relative homology, excision, Mayer-Vietoris.
  • Lecture 6. Homotopy invariance of homology.
  • Lectures 4 and 5
  • Lecture 3. Semisimplicial sets and singular homology.
  • Lecture 2: Chain complexes and homology
  • Lecture 1: Euler characteristic and categories

Answers to questions

  1. This is a pretty random question, but I was curious about the context for the Yoneda embedding; it seems like it gives us a better way to understand categories by considering the functors from it (more accurately, from its opposite category) to a category we understood better, i.e. the category of sets, kinda like how representations help us better understood a group in terms of linear transformations of something we're more familiar with, vector spaces. Wikipedia tells me that indeed, this embedding and the Yoneda lemma "underly several modern developments in algebraic geometry and representation theory." I'm not sure if this would be in the scope of the class, but what would be some key words/notable papers to search for if I wanted to learn more about these "modern developments"?

    That's a great question. The "modern developments" are probably about the developments in the last half-century. The most obvious example that springs to my mind is the "functor of points" perspective in algebraic geometry. In algebraic geometry, you study a space--just as we do in topology--but because one only considers the simplest kinds of functions on that space (algebraic functions, rather than all continuous functions) the set of functions is easy to understand. Moreover, the set of functions from a space X to a space Y is also manageable, and one often understands the space Y not just by its geometry, but often abstractly by the functor it represents, namely Hom(-,Y): AlgebraicSpaces --> Sets. This is viewing the space Y is as simply a "functor of points". This allows you to more easily think about what things like "stacks" should be. (A stack comes up when you think about a space as giving a functor into a different target category, like groupoids, or topological spaces.)

  2. What's a natural transformation good for?

    That's a great question! I can answer by philosophy, and by example.

    As an example, we'll learn about the Hurewicz map, which for instance is a map from pi_1(X) to H_1(X) for any space X. It is the map from the fundamental group to first homology, which we will prove is the abelianization of the fundamental group. But is this abelianization map compatible with maps between topological spaces? The answer is yes, and this tells us that this Hurewicz map is a natural transformation between the two functors pi_1 and H_1.

    As for philosophy, you can think of a natural transformation like a homotopy between two functors. The only difference is that this homotopy is not "invertible" in the sense that two functors F and G should not be considered similar if there is a natural transformation between them. Also, many important ideas in math are naturally defined as functors from some category C to another category D. For instance, I'll go over simplicial sets in the coming days; a simplicial set is simply a functor from a category called \Delta^op to the category of sets. And what is a morphism in the category of simplicial sets? It's most easily defined as a natural transformation between two functors.

    If you believe that morphisms are useful, then you might also believe natural transformations are useful---they are the morphisms in a category called "Functors from C to D". Another interesting perspective might be to simply think of a natural transformation as a functor C x [0,1] --> D, where C x {0} --> D is a functor F, C x {1} --> D is a functor G, and the functor C x [0,1] --> D realizes the natural transformation from F to G.