  In this paper we give a geometric construction of the Borel equivariant
(co)homology for spaces with a $G$-action, where $G$ is a compact Lie group
with the property that the adjoint representation is orientable. A nice feature
of these constructions is that there are corresponding Poincar\'e dual
(co)homology theories called backwards (co)homology. This gives rise to a third
relative (co)homology theory which we call stratifold Tate (co)homology. These
Tate groups agree with the original definition of Tate cohomology for finite
groups given by Swan. All constructions in this paper are geometric and use
stratifolds. One advantage of this description is that elements in these groups
can be described concretely by representatives. We give some examples of that.
