\(Z^k_2\)-actions fixing \(\{\text{point}\} \cup V^n\) (Q2773378)

This paper describes the equivariant cobordism classification of smooth actions \((M^m,\Phi)\) of the group \(Z_2^k\) on closed smooth manifolds \(M^m\) for which the fixed-point set of the action is the union \(F=p\smile V^n\), where \(p\) is a point and \(V^n\) is a connected manifold of dimension \(n>0\). The description is given in terms of involutions with this fixed set, i.e., the case \(k=1\). The result is applied to several special cases of \(V^n\) for which the result for involutions is known. It should be noted that the reduction to \(k=1\) is not superficial. The existence of an isolated fixed point gives a simultaneous cobordism for the normal eigenbundles of the \(Z_2^k\) action over \(V^n\).