Equivariant imbeddings of G-complexes into representation spaces (Q913319)

Let G be a finite group. We are concerned with the problem of finding a representation space V of G of minimal dimension with respect to the property that every k-dimensional G-complex X can be equivariantly imbedded in V. In case \(G=\{e\}\) we have the well-known classical result that every k-dimensional complex can be imbedded in \({\mathbb{R}}^{2k+1}\) and that this is best possible. In this paper we solve, among other things, the problem of the minimal imbedding dimension and the minimal representation space in the case of finite nilpotent groups. A somewhat simplified version of our main result is as follows. Theorem 1. Suppose G is a finite nilpotent group and that \((H_ 1),...,(H_ q)\) are G-isotropy types. Let V be a representation space of G. In order for every k-dimensional G-complex with isotropy types among \((H_ i)\), \(1\leq i\leq q\), to have an equivariant imbedding into V, the following are necessary and sufficient: (i) dim \(V^{H_ i}-\dim V^{>H_ i}\geq k+1\) and (ii) dim \(V^{H_ i}\geq 2k+1\) for \(i=1,...,q\).