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What does this naive attempt at $S^1$-equivariant homology describe?

By $G$-equivariant homology of $X$ you mean the homology of the Borel construction or homotopy orbit space $EG\times_GX$. … This "Borel homology" gives you an isomorphism for every $G$-equivariant map $X\to Y$ that is nonequivariantly a homotopy equivalence.) …
Tom Goodwillie's user avatar