Timeline for Question on coverings and and their classifying spaces
Current License: CC BY-SA 2.5
5 events
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| Mar 14, 2011 at 14:52 | comment | added | Andreas Blass |
@gadwall: They're covers, but not $|\mathbb{Z}|$-sheeted ones. Remember that homotopy classes of maps to $BG$ classify principal $G$-bundles, not any old coverings.
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| Mar 14, 2011 at 14:22 | comment | added | user13624 | This is exactly my problem. Aren't $S^2$, $S^2\times\{1,2\}$, $S^2\times\{1,2,3\}$ all covers of $S^2$? | |
| Mar 14, 2011 at 14:19 | comment | added | Andreas Blass |
$S^2\times\mathbb{Z}$ is the only cover of $S^2$. That agrees with the fact that $[S^2,S^1]$ has only one element, namely 0.
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| Mar 14, 2011 at 9:00 | comment | added | user13624 | But with this reasoning $S^2$ would have non-connected covers such as $S^2\times \mathbb{Z}$, too. How does this fit with $[S^2,S^1]=0$ but $[S^1,S^1]=\mathbb{Z}$? | |
| Mar 13, 2011 at 22:59 | history | answered | Allan Edmonds | CC BY-SA 2.5 |