A geometric proof of L\"uck's vanishing theorem for the first $L^2$-Betti number of the total space of a fibration

TL;DR

This paper provides a geometric proof of Lück's vanishing theorem for the first $L^2$-Betti number of the total space of a fibration, translating chain complex constructions into CW-complexes for better accessibility.

Contribution

It offers a new geometric proof of Lück's theorem by reformulating the original chain complex approach within CW-complexes.

Findings

First $L^2$-Betti number of the total space vanishes under certain conditions

Proof is more accessible through geometric CW-complexes

Connects algebraic and geometric methods in topology

Abstract

A significant theorem of L\"uck says that the first $L^2$-Betti number of the total space of a fibration vanishes under some conditions on the fundamental groups. The proof is based on constructions on chain complexes. In the present paper, we translate the proof into the world of CW-complexes to make it more accessible.