Summer Class on Homology

Accra, Ghana, June 2010

Class Photo on June 14, 2010. First row: Douglas Abu-Gyamfi, Margaret McIntyre. Second row: John De-Graft Mensah, Emmanuel Gazoya. Third row: Peace Ogbogbo, Thomas Katsekpor, Eyram Schwinger. Fourth row: Eric Takyi, Solomon Manukure, Daniel Ntiamoah, Daniel Kutor. Absent from picture: Agnes, Albert, Bismark, Edward Prempeh, Emmanuel Djabang, Frank Aboagye, Isaac Dontwi, Joe Jackson, Nana Akuffo, Tanko Abdullah.

Back around 1983, when I was an undergraduate student of mathematics in Tel Aviv University, I took a class in general topology and liked it very much. I remember figuring that "the next thing after" would be "algebraic topology", and I remember borrowing a book on that subject from the library. I opened that book and saw huge and hyper-complicated diagrams; much like the ones down below. And I remember feeling both fear, "how will I ever understand those?", and awe, "how on earth could this be related to properties of topological spaces and continuous functions?".

Perhaps two years later I took a class on algebraic topology, and the fear subsided and got replaced by understanding. The awe remained, though. I still find it amazing that things so abstract as "homology", "categories and functors", "commutative diagrams", and "exact sequences" together make up such a beautiful picture, that has profound implications not only in topology, but also in almost every other branch of mathematics. The awe still remains even now. These notions have become some of my everyday tools, and I almost feel comfortable with them. But at some level, I still don't understand why these tools apply so well and so often.

My plan for this class is to introduce these notions, homology, categories and functors, commutative diagrams, and exact sequences, as they appear in two particular cases. The well understood proof of the Brouwer fixed point theorem (see Wikipedia: Brouwer fixed point theorem), which uses the very basics of algebraic topology and leads to all these notions in a natural way, and the still-very-hot and still evolving subject of "Khovanov homology", which seems to suggest that these notions are even more prevalent than anybody had thought before.

I plan to cover the Brouwer fixed point theorem in the first week, and Khovanov homology in the second. The prerequisites are basic general topology (topological spaces, continuous functions, and a little about connectedness and compactness), and some basic algebra (vector spaces, Abelian groups, kernels, images, quotients). In the second week I will make some use of "the tensor product of two vector spaces".

We will meet for 3 hours a day, 5 days a week, for two weeks in June, 2010. Our approximate syllabus will be / has been:

June 14-18: The Brouwer Fixed Point Theorem		June 21-25: Knots and Khovanov Homology
Monday	Introduction, categories and functors and how the proof works, a "pictorial" definition of homology Blackboard shots and Dror's notebook.	Monday	Knots, Reidemeister moves, 3-colourings, the Kauffman bracket and some pictures and programs for the barycentric subdivision. Dror's noteboook and blackboard shots.
Tuesday	From the formal definition of homology to invariance under homotopy Dror's noteboook and blackboard shots.	Tuesday	Computations, the writhe and the Jones polynomial, preliminaries on graded spaces and tensor products. Dror's noteboook and blackboard shots.
Wednesday	Relative homology, short and long exact sequences Dror's noteboook and blackboard shots.	Wednesday	Khovanov homology for knots and links. Dror's noteboook and blackboard shots.
Thursday	Excision, the homology of spheres, homology with "small" simplices Dror's noteboook and blackboard shots.	Thursday	Aside 1: Non-Commutative Gaussian Elimination and the Rubik's Cube. Blackboard shots.
Friday	Proof of excision Margaret's blackboard, Dror's noteboook and blackboard shots.	Friday	Dessert: Hilbert's 13th Problem, in Full Colour. Blackboard shot.
Sources: Alan Hatcher's free textbook on Algebraic Topology, my 2004-05, 2005-06, and 2007-08 classes on the subject.		Sources: my papers On Khovanov's Categorification of the Jones Polynomial, Khovanov's Homology for Tangles and Cobordisms, and Fast Khovanov Homology Computations, my talk Local Khovanov Homology, and references therein.

Here's my report.