  We present a new geometric interpretation of equivariant cohomology in which
one replaces a smooth, complex $G$-variety $X$ by its associated arc space
$J_{\infty} X$, with its induced $G$-action. This not only allows us to obtain
geometric classes in equivariant cohomology of arbitrarily high degree, but
also provides more flexibility for equivariantly deforming classes and
geometrically interpreting multiplication in the equivariant cohomology ring.
Under appropriate hypotheses, we obtain explicit bijections between
$\mathbb{Z}$-bases for the equivariant cohomology rings of smooth varieties
related by an equivariant, proper birational map. We also show that
self-intersection classes can be represented as classes of contact loci, under
certain restrictions on singularities of subvarieties.
  We give several applications. Motivated by the relation between
self-intersection and contact loci, we define higher-order equivariant
multiplicities, generalizing the equivariant multiplicities of Brion and
Rossmann; these are shown to be local singularity invariants, and computed in
some cases. We also present geometric $\mathbb{Z}$-bases for the equivariant
cohomology rings of a smooth toric variety (with respect to the dense torus)
and a partial flag variety (with respect to the general linear group).
