On the torsion in the center conjecture (Q1994821)

By an almost nonnegatively curved manifold we shall understand a closed smooth manifold $M$ which admits a sequence of metrics $g_n$ with a uniform lower curvature bound such that the sequence $M_n$ converges to a point in the Gromov-Hausdorff topology. The paper under review is motivated by the following conjecture from a previous article by the authors [Ann. Math. (2) 171, No. 1, 343--373 (2010; Zbl 1192.53040)]. Conjecture 1: For all dimensions $m$ there is a constant $C = C(m)$ such that if $M^{m}$ is an almost nonnegatively curved closed smooth $m$-manifold, then there is a nilpotent subgroup $N\subset \pi_1(M^m)$ of index at most $C$ whose torsion is contained in its center. It is proved that the above Conjecture 1 (if true) implies the (Fukaya-Yamaguchi) conjecture which states that the fundamental group of a nonnegatively curved $m$-manifold $M$ is $C(m)$-abelian: there is $C = C(m)$ such that if $M^m$ is nonnegatively curved, then there is an abelian subgroup $A\subset \pi_{1}(M^m)$ of index at most $C$, cf. [\textit{K. Fukaya} and \textit{T. Yamaguchi}, ibid. (2) 136, No. 2, 253--333 (1992; Zbl 0770.53028)]. \par The main result of the paper under review proves the Conjecture 1 for some special manifolds with tower bundles. The main theorem: Let $F_1, F_2,\dots,F_n$ be an array of closed manifolds such that each $F_i$ is either ${\mathbb S}^1$ or is simply connected. Assume $E$ is the total space of a tower of fiber bundles over a point. \[ E = E_n\stackrel{F_n}\to E_{n-1}\stackrel{F_{n-1}}\to \dots \stackrel{F_1}\to E_{0} = \{pt\} \] and each of the bundles $E_k\stackrel{F_{k}}\to E_{k-1}$ is homotopically trivial over the $1$-skeleton. Then the fundamental group $\pi_{1}(E)$ contains a nilpotent subgroup $\Gamma$ such that \[ [\pi_1(E) :\Gamma]\leq \text{Const}(F_1 ,F_2,\dots,F_n) \] and Tor($\Gamma$)$\subset$Z($\Gamma$), where Tor($\Gamma$) and Z($\Gamma$) denote the torsion and the center of $\Gamma$, respectively.