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:

** **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).**
**