Math 541: Topics in Topology : Fall 2005 TTH

Professor Tim Cochran


        The class  will be a semi-standard second year graduate algebraic topology class, with more homological algebra than usual.  Along the way, I intend to have examples from knot theory and manifold theory to illustrate the utility of various notions. I also hope to have such examples from algebraic geometry if I can get a little help from class members in that area. The broad topics (not necessarily in this order) will be:


I.                   Basic Homotopy Theory


                           Homotopy groups, role of basepoint, fibrations and cofibrations, fiber bundles, long exact sequences of homotopy groups, Hurewicz Theorem, Whitehead Theorem, Eilenberg-Maclane spaces, basic obstruction theory.


II.                Homological Algebra and its applications to Homology


                          Modules (Free, projective, flat, injective), Bi-modules, Hom, Tensor product of modules, Ext, Tor, adjoint and derived functors, change of rings, LOCALIZATION OF RINGS AND MODULES, Homology with twisted coefficients, group homology.


III.             Basic Spectral Sequences


IV.              Other Topics


                     Other topics that may be covered by myself or by students in the form of a project include:  Basic theory of Bordism and Cobordism, Basic Algebraic K-theory, basic Surgery, the h-cobordism theorem, Massey products and other higher-order operations, generalized homology theories, other cohomologies (Czech, Sheaf), Postnikov towers, Thom Isomorphism theorem, Intersection theory.


                    The primary textbook will be that of James Davis and Paul Kirk, which you should purchase yourself on-line, as I have not ordered it from the Rice bookstore. There will be other sources.


                    There will be (almost) weekly required homework problems that will be collected and (sort of) graded. More advanced students can be excused from doing problems that are, for them, redundant or too easy. In addition, each student will be asked to research a special topic of interest to her/him (at least loosely related to the class) and present a 45-minute lecture to the class as well as a corresponding typed manuscript (a report or lecture notes with extra details) for the class. These will take place during the last few weeks of school  (earlier if you are ready and it is time-appropriate).