Actions of finite groups on compact manifolds
In this paper the authors address the problem of extending an action on a space Y to an action on a space X which contains Y. Several problems in transformation groups can be phrased in this way and solved using this approach. The authors reference several papers where Assadi and some other authors have proceeded in this way. Here the authors focus on the theory where X is not simply connected. Let G be a finite group of order q. A connected G-CW-complex X is called simple (and the action is called simple) if \((E_ G\times_ GX)_ q\) is fibre homotopy equivalent to \((B_ G\times X)_ q\). Here \(E_ G\) stands for the universal G space (that is, \(E_ G\) is contractible and it has a free G action) and the subscript q denotes the localization at \({\mathbb{Z}}_ q\), preserving the fundamental group, and localizing the higher homotopy groups at \({\mathbb{Z}}_ q\). The class of simple actions is rather restrictive, but in a more general situation many additional difficulties need to be solved. The main result of this paper is Theorem 2.7. Let W be a smooth compact manifold, and F a smooth submanifold, such that (F,\(\partial F)\subset (W,\partial W)\). Suppose that the codimension of F in W is at least three, and \(\pi_ 1(\partial W)=\pi_ 1(W)=\pi\), and that G acts trivially on F. Assume that the normal bundle v of F in W supports an action of G which is free off the zero section F. There exists a simple G action on W which extends the trivial action on F such that the action is free on W-F iff \[ (1)\quad H_*(W,F,{\mathbb{Z}}_ q[\pi])=0 \] \[ (2)\quad \gamma \tau (W,F)\in Wh^ T_ 1(\pi \subset \pi \times G)\quad vanishes. \] Given an action on the closed normal disk bundle, these actions are unique up to concordance. The Whitehead group and the torsion invariant in (2) are described in the paper.
scientific article; zbMATH DE number 4044771