Instructor: Stephen Bigelow.

Office: 6514 South Hall.

Office hours: M 11:00-12:30, T 1:30-3:00, or catch me after class on Friday, or send me an email, or stop by when my office door is ajar.

E-mail: bigelow@math. you know the rest

Course webpage: http://www.math.ucsb.edu/~bigelow/232a

Course description: This is an introduction to homology. The plan is to cover as much as we can of Chapter 2 of Hatcher.

Textbook: Chapter 2 of Algebraic topology by Allen Hatcher.
Also good: Munkres "Elements of algebraic topology", and Spanier "Algebraic topology".

Grading: Most of the grade will be based on regular homework assignments. Beyond that, I want to keep some flexibility. Depending on how the course progresses, I might decide to have something like a quiz, or an in class final.

Prerequisites: The prerequisites are undergraduate linear algebra and topology. It would help if you had some experience with:

• groups, rings and modules,
• cell complexes, or at least the basic idea of gluing simple things together to make an interesting topological space,
• homotopy.

Homework:

• Due 1/11: §2.1 Q1-5.
• Due 1/20: §2.1 Q11-15.
• Due 1/25: §2.1 Q16-18.

Short summary of lectures

• 1/4: A Δ-complex is made by gluing together faces of a bunch of simplices.
• 1/6: Non-rigorous calculation of homology of some examples.
• 1/8: Rigorous calculation of simplicial homology of some examples.
• 1/11: Singular homology.
• 1/13: Singular homology is a functor.
• 1/15: Homotopic functions induce identical homomorphisms on homology.
• 1/20: Relative homology.
• 1/22: A short exact sequence of chain complexes gives a long exact squence of homology.
• 1/25: The long exact sequence of relative homology.
• 1/27: Excision.
• 1/29: Relative homology vs homology of the quotient.
• 2/1: Naturality, the five-lemma, and singular vs simplicial homology.

Guest lectures

• ST: cellular homology.
• GK: Mayer-Vietoris.
• RC: Coefficients.
• CD: Axioms.
• WB: Categories.
• YC: Applications.

Web surfing

• the nLab entry on simplicial complex. As always, nLab is a good source for making distinctions like "simplicial complex" vs "simplicial set", and explaining it all in the most terrifying language possible. (Do not expect to understand anything you find at nLab.)
• Computing homology. Homology of finite simplicial complexes is effectively computable, and it looks like at least one person has actually written a program to compute it. Or, see CHomP.
• Proof of the snake lemma in a movie.
• I found a proof of excision that does not use barycentric subdivision: Acyclic models and excision by Rolf Schön. It's only two pages, but you have to know the "method of acyclic models" (roughly: do everything in Euclidean space until the last possible second).
• Hatcher is active on mathoverflow.
• If you want to do extra homework, you could do McCammond's assignments. I'd be happy to grade them.
• What I called the "snake lemma" is more accurately called the zig-zag lemma.

Short summary of the entire course

The main points of the course are:

• Definitions of the different varieties and flavors of homology.
• Proofs of the Eilenberg-Steenrod axioms.

Varieties:

• simplicial (see Munkres, or Hatcher's Δ-homology),
• singular,
• cellular.

Flavors:

• "vanilla",
• reduced,
• relative,
• different coefficients.

(Note on cellular homology: In the definition, the chain groups are singular relative homology groups, and the boundary maps are defined in the obvious way. In practice, the chain groups are integer-linear combinations of cells, and the boundary maps are computed using degrees.)

Axioms (Hatcher's version for reduced homology):

• Homotopy equivalence.
• The long exact sequence.
• Homology of wedge products.
• The homology of a point (so as not to be "extraordinary").
• (Mayer-Vietoris follows from the others).

Relative homology is similar, plus excision.

The proofs use some important lemmas:

• Chain maps induce homomorphisms on homology.
• Chain homotopic maps induce the same homomorphisms on homology.
• Zig-zag lemma.
• Five-lemma.