### WiSe 20/21: Algebraic Topology I

**Lectures:** Wednesday 10-12 and Friday 08-10 online (see the GRIPS page for the meeting link)

**Exercises:** Friday 10-12 online

Algebraic topology studies topological spaces by means of algebraic invariants (groups, vector spaces, etc.), which allow us to reduce questions in topology to questions in algebra. Algebraic topology has many applications, both in theoretical and in applied mathematics. Nowadays, a basic knowledge of algebraic topology is essential in most other fields of pure mathematics, including analysis, algebraic geometry, and number theory. In applied mathematics, topological data analysis is a relatively new field that relies heavily on tools from algebraic topology.

In this first course on algebraic topology, we will study in depth two important invariants of a topological space: its fundamental group and its (co)homology groups. We will also see how to use these algebraic invariants to answer some interesting topological questions.

Topics covered in this course include:

- Covering spaces and the fundamental group
- Simplicial sets and singular (co)homology
- CW complexes and cellular (co)homology
- Miscellaneous applications (the fundamental theorem of algebra, Brouwer’s fixed point theorem and invariance of domain, the hedgehog theorem, etc.)

This course is complemented by the seminar *de Rham cohomology*. This seminar explores another approach to the cohomology of smooth manifolds via differential forms and proves Poincaré duality, which is a fundamental homological feature of smooth manifolds not shared by more general topological spaces.

### Exercises

- Sheet 1 (due 11.11)
- Sheet 2 (due 18.11)
- Sheet 3 (due 25.11)
- Sheet 4 (due 2.12)
- Sheet 5 (due 9.12)
- Sheet 6 (due 16.12)
- Sheet 7 (due 23.12)
- Sheet 8 (due 13.1)
- Sheet 9 (due 20.1)
- Sheet 10 (due 27.1)
- Sheet 11 (due 3.2)
- Sheet 12 (due 10.2)

### Lecture notes

- §1. Homotopy
- Lecture 1 (4.11)
- Lecture 2 (6.11)
- §2. The fundamental groupoid
- Lecture 3 (11.11)
- Lecture 4 (13.11)
- §3. The Seifert–van Kampen theorem
- Lecture 5 (18.11)
- Lecture 6 (20.11)
- §4. Covering spaces
- Lecture 7 (25.11)
- Lecture 8 (27.11)
- Lecture 9 (2.12)
- Lecture 10 (4.12)
- §5. Simplicial sets
- Lecture 11 (9.12)
- Lecture 12 (11.12)
- Lecture 13 (16.12)
- §6. Singular homology
- Lecture 14 (18.12)
- Lecture 15 (23.12)
- Lecture 16 (8.1)
- Lecture 17 (13.1)
- Lecture 18 (15.1)
- §7. Cellular homology
- Lecture 19 (20.1)
- Lecture 20 (22.1)
- Lecture 21 (27.1)
- §8. Künneth, universal coefficients, cohomology
- Lecture 22 (29.1)
- Lecture 23 (3.2)
- Lecture 24 (5.2)
- Lecture 25 (10.2)
- Lecture 26 (12.2)

### References

- A. Hatcher,
*Algebraic Topology*, 2001 - C. Löh,
*Algebraic Topology, An introductory course*, Wintersemester 2018/19 - W. Lück,
*Algebraische Topologie: Homologie une Mannigfaltigkeiten*, 2005 - R. Brown,
*Topology and Groupoids*, 2006 - T. tom Dieck,
*Algebraic Topology*, 2008