Master Seminar Topology "L^2-Betti numbers"
N.N.
Additional information / Pre-requisites
Prerequisites: Basic knowledge of topology and group theory is required.
Comments
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 $H_n(X; \mathcal{N}\pi_1(X))$ with suitable coefficients.
$L^2$-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.
Detailed Information can be found on the Homepage of the seminar.
closeSuggested reading
This seminar will be an introduction to $L^2$-Betti numbers, following mostly
the book by Holger Kammeyer.
16 Class schedule
Regular appointments
Location:
A3/SR 115 (Arnimallee 3-5)
Location:
A3/SR 115 (Arnimallee 3-5)
Location:
A3/SR 115 (Arnimallee 3-5)
Location:
A3/SR 115 (Arnimallee 3-5)
Location:
A3/SR 115 (Arnimallee 3-5)
Location:
A3/SR 115 (Arnimallee 3-5)
Location:
A3/SR 115 (Arnimallee 3-5)
Location:
A3/SR 115 (Arnimallee 3-5)
Location:
A3/SR 115 (Arnimallee 3-5)
Location:
A3/SR 115 (Arnimallee 3-5)
Location:
A3/SR 115 (Arnimallee 3-5)
Location:
A3/SR 115 (Arnimallee 3-5)
Location:
A3/SR 115 (Arnimallee 3-5)
Location:
A3/SR 115 (Arnimallee 3-5)
Location:
A3/SR 115 (Arnimallee 3-5)
Location:
A3/SR 115 (Arnimallee 3-5)