\(L^ 2\)-Betti numbers of mapping tori and groups
This paper presents proofs of two conjectures of Gromov concerning \(L^2\)-Betti numbers. First, for any compact, aspherical manifold, \(M\), which fibers over the circle \(S^1\), \(b_p (M) = 0\) for each \(p\). Next, for any extension \(\{1\} \to \Delta \to \Gamma \to \pi \to \{1 \}\) of infinite groups which are fundamental groups of finite aspherical CW complexes, \(b_1 (\Gamma) = 0\). The first of these results is established as a corollary of a vanishing result for certain \(L^2\)-Betti numbers for a mapping torus \(T_f\) with \(f : F \to F\) a self map of a connected CW complex with finite skeleta through a range. Similarly, the second result comes about as a corollary of a vanishing result for \(b_1 (E)\) when \(F \to E \to B\) is a fibration of connected CW complexes with \(F\) and \(B\) having finite 2-skeleta and with suitable conditions on \(\pi_1 (F) \to \pi_1(E)\) and \(\pi_1 (B)\). For closed orientable four-manifolds, \(M\), which are geometric in the sense of Thurston, there is a complete list of all \(L^2\)-Betti numbers. In most cases they vanish. In the exceptional cases they are determined by \(|\pi_1 (M) |\) or \(\chi (M)\). It is also shown that the deficiency of any finitely presented group, \(\pi\), is bounded above by \(1 - b_0 (\pi) b_1 (\pi)\) in general, and by \(1 - b_0 (\pi) b_1 (\pi) - b_2 (\pi)\) if \(B \pi\) has.
scientific article; zbMATH DE number 590274
scientific article