Agrarian and \(\ell^2\)-Betti numbers of locally indicable groups, with a twist
type: Item
instance of: scholarly article
title: Agrarian and \(\ell^2\)-Betti numbers of locally indicable groups, with a twist
author: Dawid Kielak
author: Bin Sun
published in: Mathematische Annalen
publication date: 2024-10-28T00:00:00Z
review text: 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 p…
reviewed by: Daniel Juan Pineda
Mathematics Subject Classification ID: 57R22
Mathematics Subject Classification ID: 57K32
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 7932387
zbMATH Keywords: agrarian Betti-numbers
zbMATH Keywords: \(\ell^2\)-Betti-numbers
zbMATH Keywords: indictable groups
MaRDI profile type: Publication
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DOI: 10.1007/S00208-024-02835-7
description: scientific article; zbMATH DE number 7932387