Syllabus for Algebraic topology – Math 232A
Winter 2016

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.

## 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.

Prof. Bigelow 2016-01-01