Fixed point sets of smooth group actions on disks and Euclidean spaces (Q582642)

The author proves: Theorem A. A compact smooth manifold F (resp., a smooth manifold F without boundary) is the fixed point set of a smooth action of G on a disk (resp., Euclidean space) if and only if (1) G is a torus: F is \({\mathbb{Z}}\)-acyclic. (2) G is a 2-group or its extension by a torus: F is \({\mathbb{Z}}_ 2\)- acyclic. (3) G is a p-group or its extension by a torus for an odd prime p: F is \({\mathbb{Z}}_ p\)-acyclic and stably complex. Theorem B. Let G be either a finite group with a cyclic subgroup not of prime power order and with a normal, possibly trivial, 2-Sylow subgroup, or an extension of such a group by a torus. Then a compact smooth manifold F (resp., a smooth manifold F without boundary) is the fixed point set of a smooth action of G on a disk (resp., Euclidean space) if and only if F is a stably complex manifold with Euler characteristic \(\chi\) (F)\(\equiv 1\) (mod \(n_ G)\) (resp., F is a stably complex manifold).