Timeline for Universal covering space for non-semilocally simply connected spaces
Current License: CC BY-SA 4.0
8 events
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| Apr 25, 2020 at 14:26 | comment | added | xuan-gottfried Yang | sofar I understand the fundamental group discussed here is in reality the "instable" fundamental group. Another question of me is, could the stable fundamental group (of some good spectra) also be define by universal coverings? Or, if $E\to X$ is an universal covering space, is $\Sigma^{\infinity}E\to \Sigma^{\infinity}X$ also an universal covering object? | |
| Apr 22, 2020 at 10:37 | comment | added | YCor | But I don't think this the right place to discuss this. It seems unrelated to the question and doesn't contradict Tom's answer. You can ask a separate question to discuss whether for two extensions $L,L'$ of a field $K$, the existence of an $K$-embedding $L\to L'$ and of a $K$-embedding $L'\to L$ implies the existence of a $K$-isomorphism. | |
| Apr 22, 2020 at 10:19 | comment | added | xuan-gottfried Yang | my feeling is from algebra. Consider the category of Galois extensions of a field, say $R$, with the $R$-linear maps as morphisms where the extensions are considered as $R$- vector spaces. The projection $C \to R$ is a morphism but no field extension. I don't know any example of tow fields which were Galois extensions of each other | |
| Apr 22, 2020 at 9:36 | comment | added | YCor | (Assume $X$ deloopable, path-connected.) I'm not sure what you mean by "lifting of the universal cover": if $X$ is not simply connected there's no such lift. Tom's assertion seems correct. | |
| Apr 22, 2020 at 9:27 | history | edited | YCor | CC BY-SA 4.0 |
added reference answer
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| Apr 22, 2020 at 9:01 | history | edited | xuan-gottfried Yang | CC BY-SA 4.0 |
edited body
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| Apr 22, 2020 at 8:20 | review | Late answers | |||
| Apr 22, 2020 at 8:34 | |||||
| Apr 22, 2020 at 8:01 | history | answered | xuan-gottfried Yang | CC BY-SA 4.0 |