Math 634, Algebraic Topology

For an overview of and further information about the class, please read the syllabus. Note that homework revisions are due one week after you get back your graded homework.

For the week of 9/29:

Friday class is moved to Thursday (still at 9am in Deady 210).
For class on Wednesday, produce as many examples as you can of covering spaces of the figure eight.
For class on Thursday, we will continue to talk about maps between covering spaces and the group of deck transformations of a covering space. You are to think about covering spaces of the torus and the 2-holed torus, and you are to stare at Escher prints and think about what they have to do with covering spaces.

For the week of 10/6:

Monday - Preparation assignment: go through the proof of Theorem 1.7, which computes the fundamental group of the circle. Ask questions, make comments, and if it seems everything is clear try to adapt the proof to compute the fundamental groups of the torus, figure eight and the two-holed torus. Remember that the assignment is due (by either e-mail to dps@noether or by putting something under my door) by 8am on Monday.
For Wednesday we will talk our generalization of the computation of the fundamental group of the integers. We might start constructing the universal cover.
For Friday we will start by touching on the homotopy lifting property and then talk about the action of the fundamental group on the universal cover and its consequences for covering space theory.

For the week of 10/13:

Problem set 1 will be due Monday 10/13. The assignment is page 38 - 1, 6, 8. page 79 - 1, 2, 12, 14.
On Monday we will set up the proof of the classification theorem for covering spaces.
Wednesday - Preparation assignment: go through the classification of covering spaces, pages 63-68, culminating in Theorem 1.38. Note that you need to use the lifting criterion, Proposition 1.33 on page 61/62. Share any questions you have and/or insights into the structure of the proof.
Friday - We will introduce the Van Kampen theorem.

For the week of 10/20:

Monday and/or Wednesday - dicuss problems 4, 7 8, 14. Do not worry too much about establishing the cell structures, but feel free to use Proposition 1.26.
Friday - Preparation assignment on the proof of the Van Kampen theorem. We will meet both in the morning and in the afternoon.

For the week of 10/27:

Monday - we will continue to talk about the Van Kampen theorem.
Wednesday - we will finish the Van Kampen theorem and talk about two-dimensional CW-complexes.
Friday - to motivate homology, we will talk a bit about higher homotopy groups and see in part why they are difficult. We will come back to higher homotopy groups in the spring.
Assignment due on Monday, 11/3, from Hatcher: pg 39, problem 16 a,c,e,f and page 55, problem 22.
Preparation: Read the start of chapter two, through page 107, as we will start talking about homology on Monday 11/3.

For the week of 11/3:
This will be a transition week.

Monday - we will define higher homotopy groups and show they are abelian.
Wednesday - we will talk about cobordism and homotopy, revisit our computation of the fundamental group of the circle using degree, and try to see why the third homotopy group of the two-sphere is non-trivial. While we are having fun in class with homotopy groups, you are to be reading the introductions to homology in Hatcher's and Bredon's books.
Friday - we will define delta complexes and their homology.

For the week of 11/10:
The mid-term exam will be Monday, 11/17 in class. For review consider the following problems.

Compute the fundamental group of the two-holed torus in two different applications of the Van Kampen theorem - one by decompsing it as the union of two tori with disks removed over an annulus, and one as given by its CW-decomposition with one 0-cell, four 1-cells, and one 2-cell. Show that the groups you get with these two decompositions are isomorphic.
Show using covering space theory for S¹v S¹ v S¹ that the free group with three generators has a subgroup isomorphic to the free group with five generators. Identify the subgroup explicitly.
Give an example of a based space X and two based maps f, g : S¹ -> X, which are homotopic but not through basepoint preserving homotopy.
How would you compute the fundamental group of a delta complex?
Hatcher page 131: 1, 4, 6, 9.

Monday - we will start with the assigned computations of the delta-complex homology of the torus, Klein bottle, and projective plane. We will clarify homology of delta complexes and start to define singular homology.
Wednesday - we will discuss singular homology, its functoriality, and begin to discuss its homotopy invariance. Preparation assignment: Proposition 2.9 and Theorem 2.10 on page 111.
Friday (or Thursday?) - we will continue to dicuss homotopy invariance of singular homology and have an additional review session for the test.

For the week of 11/17:

Monday - Exam day.
Wednesday - we will finish discussion of homotopy invariance of homology.
Friday - we will introduce exact sequences, which provide the language for and computational power of the most important property/axiom of homology. To gain facility with these, the following short assignment on exact sequences is due on Wednesday, 11/26.

For the week of 11/24:

Monday - we will see how the long exact sequence of an inclusion (or pair) of spaces, in tandem with the easier properties which we have already established, allows us to compute homology. Read pages 113-top of 115, to get a head start. We will then focus on the algebraic (chain complex) version of this long exact sequence.
Wednesday - we will finish discussion of the long exact sequence in homology for a short exact sequence of chain complexes, as done in pages 116-117. We will then start to connect with topology.

The first topic will be covering after the break is excision, which is the key in constructing the long exact sequence in homology for a quotient sequence of spaces. The key to proving excision in turn is Proposition 2.21, which says roughly that homology does not change when it is made to be subordinate to an open cover. The key to this proposition is barycentric subdivision. We will break the proof of this proposition up into pieces, which you will present in groups on Wednesday and Friday of the first week of class. These parts build upon one another a bit, so the later groups will have to learn some of the material assigned to earlier groups, but this compounding of work will hopefully be mitigated by my pointing out what you can ignore from earlier in the proof (see below) and the fact that the later groups may present a full two days after the first group.

John, Dawn, Samson, Aaron: Cover section (1) of the proof, on barycentric subdivision of simplices.
Jonathan, Chad, Nathan, Timothy, Stephen: Cover section (2) of the proof, on subdivision of linear chains. Note that you can ignore the material from the first section on the diameter of the subdivision.
Seth, Bill, Michael, Lene, Max: Cover sections (3) and (4). Note that you should know the definition of the operator S on linear chains, but you can take the properties as given.

Also to prepare for lecture, everyone should give the once over to the material in the rest of the section, pages 113-130, especially the material leading up to Proposition 2.21.

Here are some review problems for the final exam, which will be on Thursday 12/11 at 10:30 am in Deady 210. Here are some more review problems.

For the week of 12/1:

Monday - no class. We will meet twice on Friday.
Wednesday - We will start with the proof of Theorem 2.16 on page 116-117. You will, in a group, be presenting part of this proof to the rest of the class.
- Max, Jonathan: define the boundary map.
- Seth, Chad, Aaron: show the boundary map is well defined.
- Dawn, Jason: show the boundary map is a homomorphism.
- Samson, Nathan: establish the first three inclusions of the proof of 2.16.
- John, Michael, Bill: Ker j_* is in Im i_*.
- Chad, Stephen, Lene: the last two inclusions of the proof.
We will go on to talk about relative homology.
Friday - we will continue to discuss the long exact sequence of a (short exact sequence of) a pair of spaces. You are to turn in the following homework, which we will review in class:
- Define CPⁿ to be the set of all non-zero vectors [z0, ..., zn] in (n+1)-dimensional complex space, quotiented by the relation [z0, ..., zn] = [c z0, ..., c zn], for any non-zero c (note that this space is defined differently in chapter 0).
  a) Show that CP¹ is homeomorphic to S², the Riemann sphere.
  b) Show that the subspace of points in CPⁿ with z0 = 0 is well-defined and homeomorphic to CP^n-1. Show that the complement to this subspace is homeomorphic to R²ⁿ.
  c) Show that the quotient space CPⁿ / CP^n-1 is homeomorphic to S²ⁿ, the one-point compactification of R²ⁿ.
  d) Compute the homology groups of CPⁿ.
  e) Repeat as much as you can of these steps for RPⁿ
- a) Read about the cone and suspension operations from chapter 0.
  b) Show that the cone on X is contractible.
  c) Show that there is an exact sequence of spaces X -> CX -> SX.
  d) Do problem 20 on page 132.
- Problem 22 on page 132.