Topics in algebraic topology | |
Student No.： | 200 |
Time： | Tue. 13:00-14:50; Thu. 8:00-9:50 am |
Instructor： | Marcel Bökstedt [Aarhus University] |
Place： | Common Room, Floor 3, Jin Chun Yuan West Building |
Starting Date： | 2012-2-28 |
Ending Date： | 2012-6-21 |
Course description:
The course "Topics in algebraic topology" will start with an introduction to simplicial sets.
A simplicial set is a somewhat complicated combinatorical structure. There are two main motivating examples:
1. For any space X the set of all maps f: n-simplex -> X form a simplicial set, the singular complex of X.
2. For any group G, the "nerve" of G is a a simplicial set.
The purpose of a simplicial set is usually to "realize" it. It's realization is a CW complex. The realization of the singular complex of X maps to X, and the map is a weak homotopy equivalence. The realization of the nerve of G is the "classifying space" BG, which is an Eilenberg- Mac Lane space K(G,1).
In his work on algebraic K-theory, Quillen realized that the construction of a nerve and a classifying space can be generalized general categories, and more surprisingly that one can actually work with this generalized definition. Today classifying spaces of categories and more general simplicial sets are everywhere in algebraic topology, and there are many examples beyond the two motivating ones.
There are several introductions to simplicial sets. One standard book is Goerss, P. G.; Jardine, J. F. (1999). Simplicial Homotopy Theory, but I don't think that I will follow any particular text in the lectures.
Suppose $G$ is a commutative group. Then there is a way of "iterating" the construction of BG, to get simplicial sets B^2G=BBG and B^3G=BBBG etc. These are also Eilenberg Mac Lane spaces, B^nG is a K(G,n). This is not so hard, given basic techniques of simplicial sets. What is more unexpected is the following. Segal discovered that if C is a monoidal category (a category with a "sufficiently commutative" product: Think of finite sets and disjoint union, or finite dimensional vector spaces and direct sum), the a similar idea gives a sequence of spaces B^nC, which form an omega-spectrum. This makes it possible to construct cohomology theories from for instance the category of finite sets, and for any field the category of finite vector spaces. The cohomology theory obtained from finite sets turns out to be stable homotopy, the cohomology theory from vector spaces is algebraic K-theory.
I will discuss Segal's famous article "Categories and cohomology theories", and give some examples beyond that (algebraic K-theory, maybe Bott periodicity).
After this, I will talk some about infinite loop spaces, probably using the book by Adams.
What I'll do after that is not quite decided yet.
Prerequisite:
Reference for the course: