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--- blog:2023-06-27 [2023/06/27 18:00] – pzhou
+++ blog:2023-06-27 [2023/06/27 18:32] (current) – pzhou
@@ Line 33: / Line 33: @@
 Consider a rank $r$ vector space over $\C$, with $\C^*$-diagonal action. First, consider the non-equivariant BM homology of the space. by definition, this is the relative homology of the one point compactification, relative to the unique point at infinity. So here, it is $H_*(S^{2r}, pt)$, which is generated by the fundamental class.
+Now, we turn on $\C^*$ or $S^1$ action. There are many invariant subspaces. We also have $S^1$ action on $S^{2r}$, coming from the suspension of $S^1$ acting on $S^{2r-1}$.
+What can we say then? If we want to compute the equivariant cohomology (which is equal to equivariant homology up to shift) of $\C$ acted by $S^1$, then we pass to the Borel model, we form $EG \times_G \C$, $G=S^1$, and we get a line bundle over $BG=\C \P^\infty$. The top dimensional 'homology', or 0-degree (degree = codim) cohomology is defined. That is $[\C_G]$, the fundamental (homology) class of the universal associated bundle.  Then, we can ask, what is the homology of $[0_G]$, the universal zero-section's fundamental class? By Thom isomorphism, we know **it is equivalent to something in the base, times the full fiber of $[\C_G] \to [0_G]$**, by perturbation and stretching. This is equal to computing the self-intersection of the zero-section.
+$$ [0_G] = [\C_G] \cdot u $$
+where $u$ is cohomological degree $2$. $ [0_G] = [\C^r_G] \cdot u^r$. This make sense at least degreewise.
+So, the actual non-compact space fundamental class is
+$$ [\C^r_G]= \frac{[0_G]}{u^r} $$
+OK, this summarizes my understanding of equivariant localization between a vector space and the zero section.
+For the abelian case, there should be several approaches, all leading to the correct consistent answers.
+BFN and VV both defined some convolution, maybe even BDG.
+==== BFN convolution space ====
+Recall what is matrix multiplication: given matrices of sizes $n_1 \times n_2$, $n_2 \times n_3$, called $M_{12}$ and $M_{23}$, we can form their product, which is given by
+$$ (M_{13})_{ij} = \sum_k (M_{12})_{ik} (M_{12})_{kj} $$
+This can be considered as composing correspondences. We need a way to do composition (multiplication), and a way to do summation (over all possible $k$, intermediate state).
+If we package this in terms of BM homology, we