A stably free nonfree module and its relevance for homotopy classification, case $Q_{28}$

Abstract

The paper constructs an “exotic” algebraic 2–complex over the generalized quaternion group of order 28, with the boundary maps given by explicit matrices over the group ring. This result depends on showing that a certain ideal of the group ring is stably free but not free. As it is not known whether the complex constructed here is geometrically realizable, this example is proposed as a suitable test object in the investigation of an open problem of C T C Wall, now referred to as the D(2)–problem.