An exotic presentation of Q_28

Abstract

We introduce a new family of presentations for the quaternion groups and show that for the quaternion group of order 28, one of these presentations has non-standard second homotopy group.

On CW-complexes over groups with periodic cohomology

“…Our proof amounts to combining recent results of W. Mannan and T. Popiel [26] with Theorem 1.1. This group was proposed as a counterexample in [2].…”

“…It should not be too difficult to replicate this proof for more examples with 4periodic cohomology and m H (G) ≥ 3, though difficulties arise in the general case. For example, to replicate this proof for all quaternion groups G = Q 4n would require a more general method of distinguishing presentations for Q 4n than the method in [26], and also require an explicit computation of the number of rank one stably free ZQ 4n -modules, extending Swan's calculations in the case n ≤ 10 [42, Theorem III].…”