Course Description -- Algebraic Topology -- Spring 2013

Instructor: Louis H. Kauffman

Office: 533 SEO

Phone: (312) 996-3066

E-mail: kauffman@uic.edu

Web page: http://www.math.uic.edu/~kauffman

Text Book: Allen Hatcher, Algebraic Topology, Cambridge University Press (2002)

Class Meetings : 12:00 noon to 12:50PM, 300TH, MWF.

Office Hours: 3PM to 4PM on MWF.

Prereqisites: Point set topology, covering spaces, fundamental group, some familiarity with homology.

Description This is a second course in algebraic topology. We will cover simplicial homology, singular homology, cohomology, cup products, Poincare duality, basics of homotopy theory and other topics as time permits.

FINAL WORKSHOP. Our final workshop/seminar is on May 9, 2013 at 3PM to 5PM in 427 SEO. Note the room change!

See ASSIGNMENT1 Here is the first assignment. It is due on Friday, January 25, 2013. Note that one more problem has been added to the set.

See ASSIGNMENT2 Here is the second assignment. It is due on Friday, February 15, 2013.

See ASSIGNMENT3 Here is the third assignment. It is due on Monday, March 4, 2013. Note: If you are frustrated by the requests to calculate homology of products of spaces in this problem set, please read pages 261-280 in Hatcher, where you will find the general theory. You can, if you wish, apply the general theory to the problems in this set. If this is your choice and you want to work on it, you can hand the problem set in later in the week. Otherwise, please calculate the first few examples (e.g. P x P at least) and see if you see the pattern of what is happening to products where there is torsion in the homology of the factors. You can also take the problem of inductively analysing the homology of P x X where X is an arbitrary topological space, in terms of the tensor product of the chains on P and the chains of X. You will find that you can write a formula for this but that your formula will involve computing a new quotient. Check this on the small examples. Note: The last problem is self-contained, but also forms the core of the theory of cohomology operations. See the Steenrod notes for more about this. The mapping Delta is, in Steenrod's terminology a chain approximation to the diagonal map (here for a simplex). A lot of topology comes from the fact that the chain approximation to the diagonal is not symmetric under the interchange of factors. That is, we have in general X -----> X x X via x ----> (x,x), but when you make a chain map that approximates the map of spaces it will not be invariant under composition with P(x,y) = (y,x).

See ASSIGNMENT4 Here is the fourth assignment. This assignment uses Dodecahedral Space and the Steenrod Lectures below.

See Steenrod. Here is a classic set of lectures by Norman Steenrod, on cup products and cohomology operations.

See Cohomology Operations. Here is an excerpt from the book "Cohomology Operations in Homotopy Theory" by Mosher and Tangora. Note how these authors are influenced by Steenrod's Lectures.

See Bredon. Chapter on products, cup products and duality from the book "Topology and Geometry" by G. Bredon.

See Products. Notes on products in cohomology from "Lectures on Algebraic Topology" by Greenberg.

See Singular Homology. Concise notes on singular homology, cohomology and duality from "Characteristic Classes" by Milnor and Stasheff.

See History. Information on the History of Algebraic Topology.

See Homotopy. Notes on homotopy theory for the last part of this course.

See Hutchings Notes. Notes on homotopy theory by Michael Hutchings.

See DeRahm. This is an excerpt about DeRahm cohomology from the book "Lecture Notes on Elementary Geometry and Topology" by Singer and Thorpe. The entire book is highly recommended reading.

See Secrets of Calculus. These are notes about calculus, differential forms and Stokes Theorem. In these notes we do infinitesimal calculus where all the infinitesimals have square equal to zero. This leads at once to Grassman algebra since 0 = (dx + dy)^2 = dx dy + dy dx + dx^2 + dy^2 = dx dy + dy dx. Infinitesimal space is non-commutative. Electrons are well aware of this, you know.

See DeRahm Notes. These are notes about DeRahm cohomology and examples.

See Fundamental Group. These are notes about fundamental group and examples from the book "Combinatorial Group Theory and Topolgy" by Stillwell.

See Axioms. A useful set of notes by John Wood on the Axioms for Homology and their consequences.