Equivariant localisation by pseudocycles

For certain groups G, acting on a manifold X, an equivariant localisation theorem relates H_*(X x_G EG) to the homology of the fixed point set of the G-action (for intuition, we use the homotopy quotient X x_G EG as an analogue of the quotient X/G when G does not act freely). This is a great tool to calculate the homology of X x_G EG, but in many cases (such as in symplectic geometry) one would like to apply localisation more generally to fibrations over some homotopy quotient. We will demonstrate that one can reinterpret the classical equivariant localisation theorem (e.g. by Atiyah-Bott or Quillen) as a pseudocycle bordism (for intuition, "almost" a bordism between two smooth embedded submanifolds). In this way, one can lift equivariant localisation from a homotopy quotient to a fibration over that homotopy quotient (given suitable conditions), and use this to perform localisation on smooth moduli spaces. Given sufficient time, we will discuss ongoing work in this area.