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Apr
7
comment When do colimits agree with homotopy colimits?
Hi Greg, perhaps you mean $X^H$ is contractible for all non-trivial subgroups $H\subset G$ ($G_+$ is cofibrant, but is not contractible for all proper subgroups unless $G$ is trivial).
Apr
6
revised $RO(G)$-graded homotopy groups vs. Mackey functors
Expanded argument to clarify dependence on rationals.
Apr
5
comment $RO(G)$-graded homotopy groups vs. Mackey functors
@AaronMazel-Gee: Sorry, that was less than clear because I rushed the answer. I filled out the answer a bit more. Hopefully, it is more understandable now.
Apr
5
revised $RO(G)$-graded homotopy groups vs. Mackey functors
Added a couple paragraphs of clarifying information and additional information for equivariant $K$-theory.
Apr
5
revised $RO(G)$-graded homotopy groups vs. Mackey functors
Added a couple paragraphs of clarifying information and additional information for equivariant $K$-theory.
Apr
5
revised $RO(G)$-graded homotopy groups vs. Mackey functors
added 1867 characters in body
Apr
4
revised $RO(G)$-graded homotopy groups vs. Mackey functors
Need suspensions to get that C_f is closed under suspension.
Apr
4
revised $RO(G)$-graded homotopy groups vs. Mackey functors
deleted 41 characters in body
Apr
4
answered $RO(G)$-graded homotopy groups vs. Mackey functors
Aug
27
awarded  Yearling
Feb
23
awarded  Nice Question
Sep
3
awarded  Enlightened
Sep
3
awarded  Nice Answer
Sep
2
answered What is the geometric fixed points of an (equivariant) Eilenberg Maclane Spectrum?
Aug
27
awarded  Yearling
Aug
20
comment How does the kernel of the map $\Omega^{\bullet}(X)\rightarrow \Omega^{\bullet}(G\times X)$ relate to equivariant cohomology?
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
comment How does the kernel of the map $\Omega^{\bullet}(X)\rightarrow \Omega^{\bullet}(G\times X)$ relate to equivariant cohomology?
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
19
revised RO(G) grading of Mackey functors
added the keyword rational
Aug
19
revised RO(G) grading of Mackey functors
added 41 characters in body
Aug
19
answered RO(G) grading of Mackey functors