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A $(G,n)$-complex is an $n$-dimensional CW-complex with fundamental group $G$and whose universal cover is $(n-1)$-connected. If $G$ has periodic cohomologythen, for appropriate $n$, we show that there is a one-to-one correspondencebetween the homotopy types of finite $(G,n)$-complexes and the orbits of thestable class of a certain projective $\mathbb{Z} G$-module under the action of$\text{Aut}(G)$. We develop techniques to compute this action explicitly anduse this to give an example where the action is non-trivial.

Projective modules and the homotopy classification of $(G,n)$-complexes