# American Institute of Mathematical Sciences

April 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. [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. [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. [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. [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. [6] T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math., 133 (1991), 317-357. doi: 10.2307/2944340.

show all references

##### 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. [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. [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. [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. [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. [6] T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math., 133 (1991), 317-357. doi: 10.2307/2944340.
 [1] Thorsten Hüls. Computing stable hierarchies of fiber bundles. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3341-3367. doi: 10.3934/dcdsb.2017140 [2] Mauro Patrão, Luiz A. B. San Martin. Morse decomposition of semiflows on fiber bundles. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 561-587. doi: 10.3934/dcds.2007.17.561 [3] Guillermo Dávila-Rascón, Yuri Vorobiev. Hamiltonian structures for projectable dynamics on symplectic fiber bundles. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1077-1088. doi: 10.3934/dcds.2013.33.1077 [4] Alberto Farina, Enrico Valdinoci. A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1139-1144. doi: 10.3934/dcds.2011.30.1139 [5] Diego Castellaneta, Alberto Farina, Enrico Valdinoci. A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983 [6] Percy Fernández-Sánchez, Jorge Mozo-Fernández, Hernán Neciosup. Dicritical nilpotent holomorphic foliations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3223-3237. doi: 10.3934/dcds.2018140 [7] Ingenuin Gasser. Modelling and simulation of a solar updraft tower. Kinetic & Related Models, 2009, 2 (1) : 191-204. doi: 10.3934/krm.2009.2.191 [8] Tracy L. Payne. Anosov automorphisms of nilpotent Lie algebras. Journal of Modern Dynamics, 2009, 3 (1) : 121-158. doi: 10.3934/jmd.2009.3.121 [9] Jordi-Lluís Figueras, Àlex Haro. Triple collisions of invariant bundles. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2069-2082. doi: 10.3934/dcdsb.2013.18.2069 [10] Bas Janssens. Infinitesimally natural principal bundles. Journal of Geometric Mechanics, 2016, 8 (2) : 199-220. doi: 10.3934/jgm.2016004 [11] V. Balaji, I. Biswas and D. S. Nagaraj. Principal bundles with parabolic structure. Electronic Research Announcements, 2001, 7: 37-44. [12] David Ginzburg and Joseph Hundley. A new tower of Rankin-Selberg integrals. Electronic Research Announcements, 2006, 12: 56-62. [13] Isaac A. García, Douglas S. Shafer. Cyclicity of a class of polynomial nilpotent center singularities. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2497-2520. doi: 10.3934/dcds.2016.36.2497 [14] Mark Pollicott. Ergodicity of stable manifolds for nilpotent extensions of Anosov flows. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 599-604. doi: 10.3934/dcds.2002.8.599 [15] Janusz Mierczyński. Averaging in random systems of nonnegative matrices. Conference Publications, 2015, 2015 (special) : 835-840. doi: 10.3934/proc.2015.0835 [16] Chaoqian Li, Yaqiang Wang, Jieyi Yi, Yaotang Li. Bounds for the spectral radius of nonnegative tensors. Journal of Industrial & Management Optimization, 2016, 12 (3) : 975-990. doi: 10.3934/jimo.2016.12.975 [17] Oliver Butterley, Carlangelo Liverani. Robustly invariant sets in fiber contracting bundle flows. Journal of Modern Dynamics, 2013, 7 (2) : 255-267. doi: 10.3934/jmd.2013.7.255 [18] Peter Albers, Jean Gutt, Doris Hein. Periodic Reeb orbits on prequantization bundles. Journal of Modern Dynamics, 2018, 12: 123-150. doi: 10.3934/jmd.2018005 [19] Andrés Contreras, Manuel del Pino. Nodal bubble-tower solutions to radial elliptic problems near criticality. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 525-539. doi: 10.3934/dcds.2006.16.525 [20] Alex Castro, Wyatt Howard, Corey Shanbrom. Complete spelling rules for the Monster tower over three-space. Journal of Geometric Mechanics, 2017, 9 (3) : 317-333. doi: 10.3934/jgm.2017013

2016 Impact Factor: 0.483

## Metrics

• HTML views (349)
• Cited by (0)

• on AIMS