Timeline for How does the kernel of the map $\Omega^{\bullet}(X)\rightarrow \Omega^{\bullet}(G\times X)$ relate to equivariant cohomology?
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| Aug 20, 2014 at 10:18 | comment | added | Justin Noel | Alternatively one can take the simplicial $G$-space $G^{\cdot+1}$ all of whose face maps are projection maps and whose degeneracies are diagonal maps. You can then take the product of this simplicial $G$-space with the constant simplicial $G$-space $X$. To obtain a simplicial $G$-space which levelwise looks the same as the one in the previous comment, but with different structure maps and different $G$-actions (this one is diagonal and realizes the Borel construction). When $X$ is $G$-free the augmentation admits an extra degeneracy and the the associated cochain complex is exact. | |
| Aug 20, 2014 at 10:05 | comment | added | Justin Noel | I don't think the projection maps will define a simplicial diagram. For example, if your diagram was simplicial, there should be an equality between the two composite face maps $(g_1,g_2,x)\mapsto g_1 g_2 x$ and $(g_1,g_2,x)\mapsto g_1 x.$ The standard construction uses the group multiplication on $G$ not the projection. Using this construction one obtains an augmented simplicial $G$ space $G^{\cdot+1}\times X\rightarrow X$. This is the bar construction on $X$ and the geometric realization is $G$-homeomorphic to the Borel construction. | |
| Aug 20, 2014 at 4:22 | history | edited | Zhaoting Wei | CC BY-SA 3.0 |
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| May 2, 2014 at 1:00 | answer | added | Pavel Safronov | timeline score: 7 | |
| May 1, 2014 at 22:50 | comment | added | Zhaoting Wei | @JohnKlein I'm considering the Cartan model. Nevertheless, any other reasonable version of equivariant cohomology is also good for me. | |
| May 1, 2014 at 22:40 | comment | added | John Klein | There are different flavors of equivariant cohomology. Which one are you considering? | |
| May 1, 2014 at 21:51 | history | asked | Zhaoting Wei | CC BY-SA 3.0 |