The course gives an introduction to algebraic topology through the theory of differential forms and de Rham cohomology groups. Among other things, we will prove the Brouwer fixed point theorem and the invariance of domain. Lectures are given in Japanese. Here is a more detailed syllabus in Japanese:

**Text:** Ib Madsen and Jørgen Tornehave: *From Calculus to
Cohomology: De Rham Cohomology and Characteristic Classes*,
Cambridge University Press, 1997.

**Time and place:** Thursday 2:45-4:15 in Science Building 1, room 509.

**Lecture notes:**

- Lecture 1: Introduction.
- Lecture 2: The alternating algebra.
- Lecture 3: The alternating algebra, continued.
- Lecture 4: Differential forms.
- Lecture 5: de Rham cohomology.
- Lecture 6: The Poincare lemma.
- Lecture 7: Cochain complexes and their cohomology.
- Lecture 8: Partition of unity.
- Lecture 9: The Mayer-Vietoris sequence.
- Lecture 10: Homotopy invariance.
- Lecture 11: Homotopy invariance, continued.
- Lecture 12: Brouwer's theorems.
- Lecture 13: The Jordan-Brouwer separation theorem.

- Problem 1. Due April 1 at 5:00 pm.
- Problem 2. Due April 7 at 5:00 pm. Solution.
- Problem 3. Due April 14 at 5:00 pm. Solution.
- Problem 4. Due April 21 at 5:00 pm. Solution.
- Problem 5. Due June 19 at 5:00 pm.
- Problem 6. Due June 25 at 5:00 pm.
- Problem 7. Due July 2 at 5:00 pm.
- Problem 8. Due July 9 at 5:00 pm.