Timeline for $\pi_0(G)$ as a subgroup of a Lie group $G$
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 30, 2020 at 17:35 | vote | accept | Ethan Dlugie | ||
| Jan 30, 2020 at 17:28 | vote | accept | Ethan Dlugie | ||
| Jan 30, 2020 at 17:35 | |||||
| Jan 30, 2020 at 8:43 | comment | added | YCor | The claim "I believe..." is not true. Consider the quaternion group $Q$ (of order 8) and $S$ the 1-dimensional circle group; let $z_Q$ and $z_S$ be their unique elements of order $2$ and $z=(z_Q,z_S)\in Q\times S$. Define $G=(Q\times S)/\langle z\rangle$. Then the quotient $C_2\times C_2$ indeed embeds into $G$, but cannot be mapped as a section (i.e., for each embedding $C_2\times C_2\to G$ there exists two distinct elements mapping into the same component). | |
| Jan 30, 2020 at 3:06 | comment | added | LSpice | See also mathoverflow.net/questions/150949/… (already served up by MO as a related question), which offers a weaker version of this property. | |
| Jan 30, 2020 at 3:04 | comment | added | LSpice | In the case where $G$ is the normaliser of a maximal torus in a reductive Lie group, your question is asking about a lift of the normaliser of the Weyl group, which does not exist in general. (The easiest example is when the ambient group is $\operatorname{SL}_2$, in which case the normaliser is a non-split extension of $\operatorname{GL}_1$ by $\mathbb Z/2\mathbb Z$.) @MikeMiller's example is the compact form of this example. | |
| Jan 30, 2020 at 2:52 | answer | added | mme | timeline score: 10 | |
| Jan 30, 2020 at 2:48 | comment | added | Olivier Bégassat | Not sure about the general case , but $\{\pm I_2\}\subset\mathrm{GL}_2(\mathbb{R})$ lie in the same component. There are two distinct questions: the existence of sections of $G\to\pi_0(G)$ and the existence of subgoups of $G$ isomorphic to $\pi_0(G)$. | |
| Jan 30, 2020 at 2:25 | review | First posts | |||
| Jan 30, 2020 at 3:42 | |||||
| Jan 30, 2020 at 2:24 | history | asked | Ethan Dlugie | CC BY-SA 4.0 |