Algebraic Topology I
The course will give an introduction to algebraic topology with a focus
on homotopy theory. We will consider the classifying space of a small
category and develop the homotopy theoretical methods used to study
its homotopy type. Later in the course, we will define and study
Here is a more detailed syllabus:
Time and place: Thursday 1:00-2:30 in 555.
- Lecture 1: The classifying space of a
- Lecture 2: The geometric realization of a
- Lecture 3: Limits and colimit; filtered
colimits and finite limits commute.
- Lecture 4: Adjoint functors, limits and
- Lecture 5: Characterizing the geometric
realization by maps from it.
- Lecture 6: Geometric realization
preserves finite products; k-spaces.
- Lecture 7: Homotopy groups; mapping
fiber; long-exact sequence.
- Lecture 8: The classifying space of a
finite group. (I recommend to skip this lecture.)
- Lecture 9: Weak equivalences, Serre
fibrations, Serre cofibrations; Quillen model categories.
- Lecture 10: The homotopy category; Quillen
- Lecture 11: Reedy model structure on
the category of simplicial spaces; geometric realization is a left
- Lecture 12: Bi-simplicial sets and their
geometric realization; geometric realization and weak equivalences;
Quillen's Theorem A and B.
- Lecture 15: Hochschild homology of a
ringoid: An introduction.