Timeline for Equivariant classifying space and manifold models
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 29, 2023 at 11:50 | comment | added | Neil Strickland | @BenWieland Yes, your suggestion is definitely correct, and well-known in the equivariant stable homotopy literature, although I am not sure what would be the most convenient reference. | |
| Mar 29, 2023 at 2:25 | comment | added | Ben Wieland | What is any other description of $B_G(S^1)$? Is it the $Y=\mathbb P(U)$, where $U$ is a complete universe (a sum of infinitely many copies of all irreps of $G$). That has a manifold approximation $\mathbb P(V)$, where $V$ is parameterized by the directed set of finite dimensional reps. You can choose a cofinal sequence if you like. This space has the property that $Y^H$ is a disjoint union of copies of $BS^1$, parameterized by homomorphisms from $H\to S^1$, which sounds right according to this. | |
| Mar 29, 2023 at 1:33 | comment | added | Jason Starr | Now I understand. A topological space $X$ with a $G$-action is equivalent to a homotopy class of a map from $(EG\times X)/G$ to $BG$. Then a $G$-equivariant circle bundle over $X$ is a lift of this map to the “relative Picard group” over $BG$ of $EG$. This is a topological covering space of $BG$ whose structure group is $H^2(G;\mathbb{Z})$ with its discrete topology. | |
| Mar 29, 2023 at 1:06 | comment | added | UVIR | @JasonStarr Here are two groups involved, a group G and $S^1$. When $G$ is trivial the answer should be $ES^1$ or $BS^1$. | |
| Mar 29, 2023 at 0:58 | comment | added | Jason Starr | The universal $G$-bundle $EG$ is a principal $G$-bundle over $BG$. | |
| Mar 29, 2023 at 0:34 | comment | added | UVIR | @JasonStarr But this is about classifying G-equivariant line bundles. The classifying space at least should have some G-action. | |
| Mar 28, 2023 at 23:33 | comment | added | Jason Starr | For every integer $m\geq 1$, for the unitary group $U(m)\subset \textbf{GL}_m(\mathbb{C})$, the classifying space is a colimit of the natural inclusions of the complex Grassmannians $\text{Grass}(m,\mathbb{C}^{m+n})\hookrightarrow \text{Grass}(m,\mathbb{C}^{m+n+1})$. So the result holds for $U(m)$. Every compact Lie group $G$ has a faithful unitary representation of finite dimension $m$, i.e., there is an embedding of compact Lie groups $G\hookrightarrow U(m)$. So the classifying space $BG$ is realized as the total space of a smooth fiber bundle over $BU(m)$ with fibers $U(m)/G$. | |
| Mar 28, 2023 at 22:29 | history | asked | UVIR | CC BY-SA 4.0 |