Equivariant SKK and vector field bordism

We use a notion of equivariant Euler characteristic in order to extend classical results on controllable cutting and pasting, and vector field bordism, to the case of manifolds acted on by an arbitrary finite group G, and modelled on a fixed virtual representation (in the sense of W. Pulikowski and C. Kosniowski). By restricting attention to such G-manifolds, one finds that classical results continue to hold in the oriented and unoriented case. This extends work of several authors.