Timeline for Loop space: De Rham cohomology
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 30, 2012 at 3:27 | comment | added | Somnath Basu | @John, eigenbunny - If $M$ is not simply connected then $LM$ is not connected; therefore, $\pi_1(LM)$ is meaningless unless you specify which component. The constant section only provides a section for $L_0M \to M$, where $L_0 M$ is the component of constant loops. The components of $LM$ are indexed by the conjugacy classes in $\pi_1(M)$. | |
| Aug 26, 2012 at 4:03 | comment | added | John Klein | @eigenbunny: Note: your short exact sequence comes equipped with a section $\pi_1(M) \to \pi_1(LM)$ induced by the inclusion of constant loops. Hence $\pi_1(LM)$ is a semi-direct product of $\pi_1(M)$ by $\pi_2(M)$. | |
| Aug 23, 2012 at 22:38 | comment | added | Mark Grant | +1 for "see by hand" :) | |
| Aug 23, 2012 at 21:09 | comment | added | Somnath Basu | It's better to say why the base point fibration (you perhaps mean the loop space fibration $\Omega M\to LM\to M$) gives a short exact sequence. In general, you get a long exact sequence; here it splits to short exact sequence as the fibration has a section. Another point to note is that $LM$ is not connected if $M$ has a fundamental group. | |
| Aug 23, 2012 at 20:27 | history | answered | eigenbunny | CC BY-SA 3.0 |