Compact Aspherical Manifolds Whose Fundamental Groups Have Nontrivial Center

Details Compact Aspherical Manifolds Whose Fundamental Groups Have Nontrivial Center Compact Aspherical Manifolds Whose Fundamental Groups Have Nontrivial Center

2011-08-11

arxiv.org/abs/1108.2321v1

Mathematics

Geometric Topology

Scientific paper

A theorem of Borel says that a closed aspherical manifold with centerless fundamental group cannot have effective action by compact connected group. Conner and Raymond further showed that only toral groups, among the connected Lie groups, can act on closed aspherical manifolds, and the fundamental group of the torus maps injectively to the fundamental group of the manifold. They further conjectured that the converse is also true. We give a counterexample to the 40 year old conjecture by constructing a closed aspherical manifold with infinite cyclic group as the center and yet admits no nontrivial circle action.

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