Math 635, Algebraic Topology


Homework due Friday 3/12: Page 157 - 21, 33. Page 165 - 1.

The final exam is on Tuesday 3/16 at 10:15 in Deady 210. It will be 3 hours long. We will have a review session on Monday at 7, discussing review problems, among other things. Your review should include the ability to clearly state the main theorems of this term.

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Syllabus synopsis:
We will continue what we have been doing towards the end of the term in MA 634, both in material and the way in which we study that material. In particular, we will be having students present material on a regular basis. We will have homeworks to be written up due approximately every other week, with a chance to make revisions after they have been graded. Added to the menu, on weeks alternating with the written-up homeworks, we will assign problems which we will discuss during our time slot on Friday afternoon. We will have a mid-term and a final exam; grading will be as it was last term.


Week zero:
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. 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). 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.


Week 1, 1/5-1/9:

• On Monday we will introduce relative homology and state excision.
• On Wednesday and Friday you will present barycentric subdivision, as needed to prove excision.

Week 2, 1/12-1/16:
Homework due Friday, 1/14: pages 132-133, 16(a), 17, 21, 25, 27.

• On Monday we will finish your presentation of the proof that the chains subordinate to some open cover have the same homology as the full chain complex.
• On Wednesday we will use the subordination of chains to an open cover to prove excision and (finally) the fact that a "short exact sequence of spaces" gives rise to a long exact sequence in homology. Remind yourself of the set-up to Proposition 2.21 and then read pages 124-125. We will not do Example 2.23 in class, but it going through it carefully will be good review of the concepts. We may also start to talk about naturality.
• On Friday we will prove the equivalence of singular and simplicial homology. This will be our first "five-lemma argument". To prepare, read the statement of the lemma and try to prove it yourself - it's a fun game!
  We will probably not meet during our afternoon slot on Friday.

Weeks 3-5, 1/21-2/6:

After being singularly focused (pardon the pun) on proving excision and developing the long exact sequence associated to a short exact sequence of spaces over the past three weeks, we are going to branch out in a couple directions in parallel now. One will be to develop the Mayer-Vietoris sequence, a close cousin to our standard long exact sequence, and use it for both computational and theoretical applications. On the computational end, we will have some discussion problems for Friday afternoon, 1/29. Our other direction will be to develop the degree of a map from both the manifold and homology perspectives, which will include reading from Milnor's lovely little book on this subject. This will be the first place we see manifold theory and algebraic topology interact. You will be presenting some of this material.
• Wednesday, 1/21: I will finish the five-lemma proof that simplicial and singular homology are isomorphic (pages 125 and 128-131 in the book).
• Friday, 1/23 and Monday, 1/26: I will introduce the Mayer-Vietoris sequence, the analogue in homology of the Seifert-Van Kampen theorem. We will use it to make computations, construct examples and, along with the Whitehead theorem - which we will not prove - to establish homotopy equivalence. See pages 149-151 in the text. One application will be to "representing and killing classes", which is not in Hatcher's book.
• Wednesday, 1/28 and Friday 1/30: We will switch gears and talk a bit about manifolds and degree of a map. There will be two developements of degree. We will use Milnor's "Topology from the Differentiable Viewpoint" for the manifold-theoretic development.
  • I will cover the basics on manifolds (which in this book are submanifolds living in the womb of Euclidean space), pages 1-7. This might start on Monday.
  • Max and Timothy will talk about regular values, page 8, presenting the definition and giving examples, of their own making, of how the definitions work for a map from R¹ to R¹, from R¹ to R², from R² to R¹ and from R² to R² (for example f(x,y) = (x² + y², x²-y²)).
  • You can read a proof of the fundamental theorem of algebra on your own for fun; it will not be part of class.
  • Dawn, Chad and Bill will present Sard's and Brown's theorems, pages 10 and 11. Just state Sard's theorem but go ahead and prove Brown's from it. Give examples as above (in fact, preferably you can use the same functions as Max and Tim).
  • The Brouwer fixed point theorem and the proof of Sard's theorem may be read for your own enrichment, but will not be presented.
  • Lene, John and Seth present Mod 2 Degree, pages 20-25. Be very brief about smooth homotopy. Prove the homotopy lemma. Only state the homogeneity lemma. Prove the main theorem on page 24, and give examples as from the book.
  • Sammy, Stephen and Aaron present Oriented Manifolds and Brouwer Degree, pages 26-31. Do pretty much everything, though don't worry if you do not do all of the examples.
  • Finally, Michael, Jonathan and Nathan will present degree between maps of spheres from the homological viewpoint out of Hatcher, pages 134-137. They should do their best, perhaps talking with me some, to prove that this is the same as the Brouwer degree.
• Friday 1/30 afternoon: discussion of some Mayer-Vietoris problems, namely problems 28,29 and 36 on pages 157-158 of Hatcher's book.

Week 6:
We will have an exam on Friday, February 13th. Here are some review problems.
Some answers, which we can discuss during review on Thursday 2/12 at 1:
6) The reduced homology of real projective space is 0 in even degrees, Z/2 in odd degrees, except if the dimension of the projective space is odd, in which case the top degree has homology Z.
7) Z^n-1 in degree 2, Z²ⁿ in degree 1.
8) Z² in degree 2, Z in degree 1.

Week 7, 2/16-2/20:

We will be developing and applying cellular homology.
• On Monday we will review the construction of CW complexes and give the definition of the cellular chain complex, doing some of the simplest examples.
• On Wednesday and Friday we will go through some of the basic examples of spaces whose homology we can compute cellularly: Sⁿ, surfaces, RPⁿ, (Sⁿ)^k, CPⁿ, Moore spaces. Once we are done with these basic examples, we will start on proving that cellular homology is well-defined and agrees with singular homology.
• On Friday afternoon we will talk about example 2.38 on page 143 and exercises 11 and 13 on page 156.

Week 8, 2/23-2/27: We will show that the cellular chain complex computes the homology of a CW-complex. The main topics over the next few weeks will be: introduction of the language of categories and functors, Euler characteristic, the Lefschetz fixed point theorem, and homology with coefficients.

Homework due Monday, March 1: page 156 - 10, 13, 17 (note that you need to review naturality from page 127, which was not treated in class), 24. We will discuss these problems on Friday afternoon.

Week nine:
We will talk about categories and functors, both in how they are a useful language for describing the basic methods in algebraic topology (algebraic functors being used to capture geometric situations) and how they are a flexible tool, especially when considering functors from "really" small categories. We will also show that familiar constructions such as groups can be framed in terms of categories, as can (nice) spaces - and conversely that to any small category one can associate a canonical CW-complex. Read pages 160-165 of Hatcher.

• Monday - introduction to categories + many examples.
• Wednesday - introduction to functors + many examples + focus on homology as a functor.
• Friday - introduction to natural transformations + the classifying space of a category (which we use to shake our world up in many ways).

Week ten:
We will develop the Euler characteristic and the Lefshetz number (two concepts which look much alike, and are in fact related). Euler characteristic will be on Monday (see page 146), followed by an introduction to Lefshetz number, which will then be developed over the rest of the week.
The Lefshetz theorem is on pages 179-182. Please read through this closely before class on Wednesday. You may also submit questions before class, to help guide class time.