Seminar: Algebraic Topology (SS 2021)
The content of this page can also be found in the corresponding OLAT course. The virtual classroom used for the seminar can only be accessed from this course.
- Time: Fri 14:00-15:30, starting April 23
If you would like to participate in the seminar, please write me an e-mail (firstname.lastname@example.org) as soon as possible.
Content of the Seminar
From the “Introduction to Topology” class you know already the fundamental group π1(X) of a topological space X. It measures whether there are non-contractible loops in X, i.e. whether X contains certain types of “holes”.
It is the general idea of algebraic topology to assign to topological spaces various algebraic objects (groups, rings, vector spaces) that detect to what extent X is “topologically non-trivial”. One can then study topological spaces with algebraic methods, e.g. prove that two spaces are not homeomorphic by exhibiting an associated algebraic object that is not isomorphic for the two spaces.
The fundamental group is the easiest algebraic object of this type. However, it has several drawbacks: It is often hard to compute, the resulting group is quite complicated even for reasonably simple spaces, and last but not least it is unable to detect many topological properties. For example, the 2-sphere is certainly not contractible, although its fundamental group is trivial.
It is therefore the goal of this seminar to introduce the so-called homology and cohomology groups that do not have these disadvantages. They are an indispensable tool in the study of “everyday topological spaces” such as manifolds or other reasonably well-behaved subsets of ℝn (with the standard topology). As the main objects of algebraic geometry, namely complex algebraic varieties, are of this type, this seminar is particularly useful for students who intend to specialize in the area of algebraic geometry.
There are many applications of algebraic topology that we will discuss, often providing easy rigorous proofs of intuitive statements that would be virtually impossible to show with other methods. For example:
- ℝn and ℝm are not homeomorphic if n ≠ m .
- Brouwer's Fixed Point Theorem: any continuous map from an n-dimensional ball to itself has a fixed point.
- The “Hairy Ball Theorem”: you cannot comb a hedgehog without a bald spot.
- The Jordan Curve Theorem: every closed curve in ℝ2 separates the plane into (exactly) two connected components.
In addition, we will see that algebraic topology can often be used for counting purposes, e.g. to determine the number of inverse images of a given point under a continuous map – vastly generalizing the fundamental theorem of algebra that a complex polynomial of degree n has exactly n zeroes (when counted with multiplicities).
The content of the classes “Algebraic Structures” and “Introduction to Topology" will be assumed. Knowledge of some parts of the “Commutative Algebra” class (regarding modules and exact sequences) will be useful.
List of Talks
Here is a list of talks in the seminar. The references point to Hatcher's book mentioned below, and are always meant so that only selected parts of them are covered in the talk. Please contact me well in advance to discuss your talk!
- May 21: Degree and Cellular Homology (Chapter 2.2 up to Theorem 2.35)
- May 28: Applications of Cellular Homology (page 140 to top of page 153)
- June 4: Classical Applications (Chapter 2.B)
- June 11: The Universal Coefficient Theorem (Chapter 3 up to page 197)
- June 18: Singular Cohomology (page 197 to end of Chapter 3.1)
- June 25: Cup Products (Chapter 3.2 up to page 214)
- July 2: The Künneth Formula (bottom of page 214 to page 222)
- July 9: Orientations (Chapter 3.3 up to Proposition 3.29)
The seminar will (roughly) follow the main parts of Chapters 2 and 3 in the book “Algebraic Topology” by Allen Hatcher.