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On the torsion in the center conjecture. (English) Zbl 1412.57018

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.

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
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References:

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[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
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