On the torsion in the center conjecture

TL;DR

This paper establishes a condition for fiber bundle towers ensuring the fundamental group of the total space has a finite-index nilpotent subgroup with torsion in its center, motivated by a conjecture on almost nonnegatively curved manifolds.

Contribution

It provides a new criterion linking fiber bundle structures to the algebraic properties of the fundamental group, advancing understanding of geometric group theory.

Findings

Identifies conditions under which the fundamental group has a nilpotent subgroup with torsion in the center.

Bounds the index of the subgroup in terms of the fibers.

Supports the conjecture relating curvature, fiber structure, and fundamental group properties.

Abstract

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 M' for which the number of leafs is bounded in terms of m such that the torsion of the fundamental group of M' lies in its center.