  Let G be a locally compact group, let X be a universal proper G-space, and
let Z be a G-equivariant compactification of X that is H-equivariantly
contractible for each compact subgroup H of G. Let W be the resulting boundary.
Assuming the Baum-Connes conjecture for G with coefficients C and C(W), we
construct an exact sequence that computes the map on K-theory induced by the
embedding of the reduced group C*-algebra of G into the crossed product of G by
C(W). This exact sequence involves the equivariant Euler characteristic of X,
which we study using an abstract notion of Poincare duality in bivariant
K-theory. As a consequence, if G is torsion-free and the Euler characteristic
of the orbit space X/G is non-zero, then the unit element of the boundary
crossed product is a torsion element whose order is equal to the absolute value
of the Euler characteristic of X/G. Furthermore, we get a new proof of a
theorem of Lueck and Rosenberg concerning the class of the de Rham operator in
equivariant K-homology.
