# American Institute of Mathematical Sciences

2018, 25: 27-35. doi: 10.3934/era.2018.25.004

## On the torsion in the center conjecture

 1 Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada 2 Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, USA 3 Wilderich Tuschmann, Arbeitsgruppe Differentialgeometrie, Institut für Algebra und Geometrie, Fakultät für Mathematik, Karlsruher Institut für Technologie, Englerstr. 2, D-76131 Karlsruhe, Deutschland

Received  July 31, 2017 Published  April 2018

We present a condition for towers of fiber bundles which implies that the fundamental group of the total space has a nilpotent subgroup of finite index whose torsion is contained in its center. Moreover, the index of the subgroup can be bounded in terms of the fibers of the tower.

Our result is motivated by the conjecture that every almost nonnegatively curved closed $m$-dimensional manifold $M$ admits a finite cover $\tilde M$ for which the number of leafs is bounded in terms of $m$ such that the torsion of the fundamental group $π_1 \tilde M$ lies in its center.

Citation: Vitali Kapovitch, Anton Petrunin, Wilderich Tuschmann. On the torsion in the center conjecture. Electronic Research Announcements, 2018, 25: 27-35. doi: 10.3934/era.2018.25.004
##### References:
 [1] J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2), 96 (1972), 413-443.  doi: 10.2307/1970819.  Google Scholar [2] E. Dror, W. G. Dwyer and D. M. Kan, Self-homotopy equivalences of virtually nilpotent spaces, Comment. Math. Helv., 56 (1981), 599-614.  doi: 10.1007/BF02566229.  Google Scholar [3] K. Fukaya and T. Yamaguchi, The fundamental groups of almost nonnegatively curved manifolds, Annals of Math. (2), 136 (1992), 253-333.  doi: 10.2307/2946606.  Google Scholar [4] V. Kapovitch, A. Petrunin and W. Tuschmann, Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces, Annals of Math., 171 (2010), 343-373.  doi: 10.4007/annals.2010.171.343.  Google Scholar [5] B. Wilking, On fundamental groups of manifolds of nonnegative curvature, Differential Geom. Appl., 13 (2000), 129-165.  doi: 10.1016/S0926-2245(00)00030-9.  Google Scholar [6] T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math., 133 (1991), 317-357.  doi: 10.2307/2944340.  Google Scholar

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##### References:
 [1] J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2), 96 (1972), 413-443.  doi: 10.2307/1970819.  Google Scholar [2] E. Dror, W. G. Dwyer and D. M. Kan, Self-homotopy equivalences of virtually nilpotent spaces, Comment. Math. Helv., 56 (1981), 599-614.  doi: 10.1007/BF02566229.  Google Scholar [3] K. Fukaya and T. Yamaguchi, The fundamental groups of almost nonnegatively curved manifolds, Annals of Math. (2), 136 (1992), 253-333.  doi: 10.2307/2946606.  Google Scholar [4] V. Kapovitch, A. Petrunin and W. Tuschmann, Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces, Annals of Math., 171 (2010), 343-373.  doi: 10.4007/annals.2010.171.343.  Google Scholar [5] B. Wilking, On fundamental groups of manifolds of nonnegative curvature, Differential Geom. Appl., 13 (2000), 129-165.  doi: 10.1016/S0926-2245(00)00030-9.  Google Scholar [6] T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math., 133 (1991), 317-357.  doi: 10.2307/2944340.  Google Scholar
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