Let $X$ be a manifold, acted on by a Lie group $G$. (For example $X$ real-even-dimensional acted on by $G=U(1)$ with only finitely many isolated fixed points.) The Cartan model for $G$-equivariant cohomology of $X$ with real coeffificents is built using the differential $d_e := d - i_v$, where $v \in Lie G$ is a vector field and $i$ is contraction.
Is there a similar construction for equivariant homology, where one introduces a boundary operator of the form $\partial_e = \partial - j$ (for some operator $j$) and writes an equivariant cycle in terms of usual cycles multiplied by polynomials of some Lie algebra parameter? I'm looking for a construction that is equivalent to Borel-Moore equivariant homology, in the same spirit as Cartan model above is equivalent to Weyl model of equivariant cohomology.
