turkmath.org

I will try to discuss a few more or less classical topics that I always wanted to cover in my algebraic topology courses but never managed to find time for. We will start with the topological Riemann--Roch theorem and Thom's approach to the construction of the Stiefel--Whitney classes. As an immediate consequence, we will obtain Wu formulas, the homotopy invariance of the characteristic classes of smooth manifolds, universal relations between characteristic classes/numbers of manifolds, etc. Next, I will illustrate how the same techniques applied to other extraordinary cohomology theories may yield such deep results as, say, the divisibility theorems for Pontrjagin numbers. Alternatively, if time permits, I will try to discuss an equivariant version of Wu classes in real algebraic varieties.