19223811 Masterseminar Topologie "L^2-Betti numbers"
- FU-Students should register via Campus Management.
- Non-FU-students should register via MyCampus/Whiteboard.
Winter Term 2025/2026
Dozenten: Dr. Kevin Li, Prof. Dr. Elmar Vogt
Time and place: Thursdays, 10am -- 12am, SR 115, Arnimallee 3.
Examination consists of two parts:
- Talk of ca. 80 min (blackboard, not slides).
Fix a meeting with us for approx. 2 weeks before your scheduled talk to discuss questions. - Written report of ca. 10 pages (Latex, not handwritten).
Deadline for report: 05.02.2026
Schedule
The detailed program with talk descriptions can be found here.
Talks 11 and 12 are still available. Please contact us if you would like to give one of them.
| Date | Title | Speaker |
|---|---|---|
| 16.10. | Organisatorial meeting: Overview and distribution of talks | Kevin Li, Elmar Vogt |
| 23.10. | Talk 1: The group von Neumann algebra | Chris Huggle |
| 30.10. | Talk 2: The von Neumann dimension | Elia Auer |
| 06.11. | Talk 3: $G$-CW-complexes | Valentina Taylor Cerra |
| 13.11. | Talk 4: $L^2$-Betti numbers | Ruochong Huang |
| 20.11. | Talk cancelled | N.N. |
| 27.11. | Talk 6: Applications to topology | Laura Weyers |
| 04.12. | Talk 7: $L^2$-Betti numbers of general $G$-spaces | Kevin Li |
| 11.12. | Talk 8: Applications to group theory | Cai Longbo |
| 18.12. | Talk 5: Applications to open problems | Elmar Vogt |
| 2026 | ||
| 08.01. | Talk 9: Lück’s approximation theorem: Statement and functional calculus | Abhijeet Vats |
| 15.01. | Talk 10: Lück’s approximation theorem: Proof and extensions | Abhijeet Vats |
| 22.01. | Talk 11: Rank gradient and cost | N.N. |
| 29.01. | Talk 12: The approximation conjecture | N.N. |
| 05.02. | Deadline for handing in report | everyone |
| 12.02. |
Description
Prerequisites: Basic knowledge of topology and group theory is required.
Content: The Euler characteristic of finite CW-complexes is multiplicative under finite-sheeted coverings and it is homotopy invariant. These properties can be deduced from different descriptions:
1. As the alternating sum of the numbers of cells, which are multiplicative but not homotopy invariant.
2. As the alternating sum of Betti numbers, which are homotopy invariant but not multiplicative. The $n$-th Betti number of $X$ is the $\mathbb{Q}$-dimension of the homology $H_n(X;\mathbb{Q})$ with rational coefficients.
3. As the alternating sum of $L^2$-Betti numbers, which enjoy the best features from both worlds: they are multiplicative and homotopy invariant. The $n$-th $L^2$-Betti number of $X$ is the von Neumann-dimension of the homology $H_n(X; \mathcal{R}(\pi_1X))$ with twisted coefficients in the group von Neumann algebra.
$L^2$-Betti numbers are meaningful topological invariants, as they obstruct the structures of mapping tori and $S^1$-actions. They also have applications to group theory by considering the $L^2$-Betti numbers of classifying spaces. Moreover, $L^2$-Betti numbers are related to famous open problems, such as the Hopf and Singer conjectures on the Euler characteristic of manifolds, and the Kaplansky conjecture on zero divisors in group rings.
References: We will mostly follow the book ``Introduction to $\ell^2$-invariants" by Holger Kammeyer.