Agrarian and \(\ell^2\)-Betti numbers of locally indicable groups, with a twist (Q6624819)

An old conjecture by Hopf relates the curvature of an even dimensional Riemannian manifold and its Euler characteristic. This evolved over the years and was reformulated by Thurston in the following way: Let \(M\) be a \(2n\)-dimensional aspherical manifold, then \((-1)^n\chi(M)\geq 0\), where \(\chi(M)\) denotes the Euler characteristic of \(M\). A method to approach this problem was used by Singer by using \(\ell^2\)-Betti numbers and then compare them to the usual Betti numbers via the \(L^2\) Index Theorem of Atiyah. In this work, the authors estimate the \(\ell^2\)-Betti numbers in case of specific fibrations, \(F\to E\to B\), promoting knowledge of \(\ell^2\)-Betti numbers of \(B\) and properties if \(F\) to estimate \(\ell^2\)-Betti numbers of \(E\). A group \(G\) is \textit{locally indicable} if every non-trivial finitely generated subgroup admits an epimorphism onto \(\mathbb{Z}\). Given a space \(X\), denote by \(b_i^{(2)}(X)\) and \(b_i(X)\) the ith-\(\ell^2\) and ith-regular Betti numbers, respectively. The main theorem is the following \N\N\textbf{ Theorem 1.1 } Let \(F\to E \to B\) be a fibration of connected finite \(CW\)-complexes. Assume \(\pi_1(B)\) is virtually locally indicable. If the induced map \(\pi_1(E)\to \pi_1(B)\) is an isomorphism or \(F\) is simply connected then \N\[\Nb_i^{(2)}(E)\leq \sum_{j=0}^{i}b_j(F)\cdot b^{(2)}_{i-j}(B) \text{ for every } i\in\mathbb{N}. \N\]\NFurthermore, if the homology of \(F\) with \(\mathbb{C}\)-coefficients is non-zero in at most two degrees, \(0\) and \(n\) with \(n\geq \text{max}\{2,\text{dim }B\}\), then for every \(i\in \mathbb{N}\) we have \N\[\Nb^{(2)}_i(E)=b^{(2)}_i(B)+b_n(F)\cdot b^{(2)}_{i-n}(B). \N\]\NThe authors include applications of the above when \(B\) is a compact connected surface different from the sphere, the projective plane and the 2-disk and to the case when \(B\) is an orientable prime 3-manifold with empty boundary or toroidal boundary and infinite fundamental group. In the second part of the paper, the authors compare the twisted Alexander and the Thurston norms on \(H^1(G;\mathbb{R})\) for groups that are either \textit{free-by-cyclic} or fundamental groups of closed connected orientable 3-manifolds that fiber over the circle.