Kapovitch, Vitali; Petrunin, Anton; Tuschmann, Wilderich On the torsion in the center conjecture. (English) Zbl 1412.57018 Electron. Res. Announc. Math. Sci. 25, 27-35 (2018). 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. [K. Fukaya and T. Yamaguchi, ibid. (2) 136, No. 2, 253–333 (1992; Zbl 0770.53028)]. 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. Reviewer: Andrzej Szczepański (Gdańsk) MSC: 57R19 Algebraic topology on manifolds and differential topology 55R10 Fiber bundles in algebraic topology 57M05 Fundamental group, presentations, free differential calculus 53C20 Global Riemannian geometry, including pinching Keywords:nonnegative curvature; nilpotent; tower of fiber bundles Citations:Zbl 1192.53040; Zbl 0770.53028 PDFBibTeX XMLCite \textit{V. Kapovitch} et al., Electron. Res. Announc. Math. Sci. 25, 27--35 (2018; Zbl 1412.57018) Full Text: DOI arXiv References: [1] J. Cheeger; D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2), 96, 413-443 (1972) · Zbl 0246.53049 · doi:10.2307/1970819 [2] E. Dror; W. G. Dwyer; D. M. Kan, Self-homotopy equivalences of virtually nilpotent spaces, Comment. Math. Helv., 56, 599-614 (1981) · Zbl 0504.55004 · doi:10.1007/BF02566229 [3] K. Fukaya; T. Yamaguchi, The fundamental groups of almost nonnegatively curved manifolds, Annals of Math. (2), 136, 253-333 (1992) · Zbl 0770.53028 · doi:10.2307/2946606 [4] V. Kapovitch; A. Petrunin; W. Tuschmann, Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces, Annals of Math., 171, 343-373 (2010) · Zbl 1192.53040 · doi:10.4007/annals.2010.171.343 [5] B. Wilking, On fundamental groups of manifolds of nonnegative curvature, Differential Geom. Appl., 13, 129-165 (2000) · Zbl 0993.53018 · doi:10.1016/S0926-2245(00)00030-9 [6] T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math., 133, 317-357 (1991) · Zbl 0737.53041 · doi:10.2307/2944340 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.