Math 795 Course Description

Igor Kriz

I will cover topics which belong to algebraic topology, and have broader use, but usually don't get covered by 695 or 696. For example, this includes spectral sequences, simplicial sets, Postnikov towers, localization and completion of spaces, generalized cohomology theory, equivariant topology, rational homotopy theory, cohomology operations, and others.

The selection of topics will be motivated by certain geometrical questions, which include some of the best results of modern algebraic topology.

• Why are there no non-zero maps (up to homotopy) from the infinite real projective space to any finite simplicial complex?
• Why is the K-theory of BG for a compact Lie group G the completion of the representation ring of G?
• How can one calculate in practice the torsion-free part of the homotopy groups of any space?
• Why is the 0-th stable cohomotopy group of BG for a finite group G a completion of its Burnside ring (the Grothendieck group of finite G-sets under disjoint sum)?

The course will have more the form of a ``leisurely stroll'' than rush toward any particular point. Its real goal is for the audience to learn advanced techniques needed for research in algebraic topology and adjacent areas (algebraic geometry, geometric topology, representation theory).

There are no formal prerequisites for this course, but I would like to assume that the audience knows the material of 695, or its equivalent.