Linear systems over [formula omitted] and roots of maps of some 3-complexes into [formula omitted

Abstract

Let Z[Q32] be the group ring where Q32=〈x,y|x8=y2,xyx=y〉 is the quaternion group of order 32 and ε the augmentation map. We show that, if PX=K(x−1) and PX=K(−xy+1) has solution over Z[Q32] and all m×m minors of ε(P) are relatively prime, then the linear system PX=K has a solution over Z[Q32]. As a consequence of such results, we show that there is no map f:W→MQ32 that is strongly surjective, i.e., such that MR[f,a]=min⁡{#(g−1(a))|g∈[f]}≠0. Here, MQ32 is the orbit space of the 3-sphere S3 with respect to the action of Q32 determined by the inclusion Q32⊆S3 and W is a CW-complex of dimension 3 with H3(W;Z)=0.