THE EQUIVARIANT EULER CHARACTERISTIC

We describe an equivariant version of the Euler characteristic in order to extend to the equivariant case classical results relating the Euler characteristic to vector field (Reinhart) bordism of smooth manifolds and controllable cut-and-paste equivalence. We show that the nonequivariant results continue to hold for an arbitrary finite ambient group G, both in the oriented and unoriented cases, and thereby extend work on this subject begun by several authors. We use a new definition of equivariant orientation in terms of a categorical notion of 'groupoid representations'.