On \(L^ 2\)-homology and asphericity
The paper describes the use of \(L^2\)-homology and \(L^2\)-Betti numbers for questions about deficiency and Euler characteristics of finitely presented groups. For instance, if \( G\) is a finitely presented group with deficiency one and can be written as an extension of \(\mathbb{Z} \) by a finitely generated normal subgroup, then the 2-complex associated to any presentation of optimal deficiency of \(G\) is aspherical. If \(X\) is an \(m\)-dimensional CW-complex with fundamental group \(\pi\) whose universal covering is \((m-1)\)-connected, then \(X\) is aspherical if and only if \(\chi (X)=0\). Applications to the Whitehead conjecture, 4-manifolds and 2-knot groups are discussed.
scientific article; zbMATH DE number 1062235