The divisibility in the cut-and-paste group of $G$-manifolds and fibring over the circle within a cobordism class

Abstract

We prove a divisibility theorem for elements in the cut-and-paste group, or the $SK$-group of $G$-manifolds, $G$ a finite abelian group of odd order. As an application we obtain necessary and sufficient conditions for that a closed $G$-manifold is equivariantly cobordant to the total space of $G$-fibration over the circle.