The authors prove, first, the following result (previously announced in print) giving necessary and sufficient conditions for the existence of an action of a \(p\)-toral group (i.e. a finite \(p\)-group or an extension of a finite \(p\)-group by a torus) on a disk (or Euclidean space) with a given manifold fixed point set:
Theorem A. Let \(G\) be a \(p\)-toral group and let \(M\) be a compact (respectively, open) smooth manifold. Then a smooth action of \(G\) on a disk (respectively, Euclidean space) \(E\) such that \(\mathrm{Fix}^G(E)\) is diffeomorphic to \(M\) exists if and only if \(M\) is both \(\mathbb{F}_p\)-acyclic and stably complex.
The authors’ proof, which fixes a gap in the literature, is based on the use of \(G\)-equivariant \(K\)-theory to solve a \(G\)-vector bundle extension problem, along with \(G\)-equivariant thickening techniques used to turn a \(G\)-CW complex into a smooth \(G\)-manifold. These tools also yield a proof of the following generalization of Theorem A, corresponding to the second main result in the article:
Theorem B. Let \(G\) be a \(p\)-toral group and let \(M\) be a compact (respectively, open) smooth manifold. Let \(K\) be a finite \(\mathbb{Z}\)-acyclic CW complex admitting a cellular map of period \(p\), with exactly one fixed point. Then a smooth action of \(G\) on a compact (respectively, open) smooth manifold \(E\) of the homotopy type of \(K\) such that \(\mathrm{Fix}^G(E)\) is diffeomorphic to \(M\) exists if and only if \(M\) is both \(\mathbb{F}_p\)-acyclic and stably complex.