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

However, a few simple examples show that $\Delta$-cohomology and equivariant homology are certainly not equal. … Is it related to $S^1$-equivariant homology in any way? If this is not possible in the general case, is it at least possible for $X$ of the form $LM$, with $M$ a manifold? …