Timeline for Fundamental groups between surjective proper fibrations
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Jul 29, 2023 at 20:29 | comment | added | Ben C | @higgs-boson Fix a connected CW complex $X$. Taking the fiber over a fixed base-point induces an equivalence of categories between covering spaces of $X$ and sets with a $\pi_1$-action where the connected ones correspond to sets with a transitive action (and hence to subgroups of $\pi_1$). This will be spelled out in any algebraic topology book (sometimes going under the name "galois correspondence"). Then use the elementary group theory fact: $G \to H$ is surjective if and only if every transitive $H$-set is also transitive when viewed as a $G$-set. | |
| Jul 29, 2023 at 9:49 | comment | added | Higgs-Boson | @ David White: thanks! I am familiar with Hatcher's book. But I did not find the theorem "A map of connected CW complexes induces a surjection on $\pi_1$ if and only if the pullback of any connected covering space is connected. " Do you know how to prove it? | |
| Jul 29, 2023 at 6:08 | comment | added | David White | A reference is Allen Hatcher's book, free online. Go to the section on covering spaces | |
| Jul 29, 2023 at 5:45 | comment | added | Higgs-Boson | @ Ben C. Thanks a lot! Could you please give a reference of your theorem? | |
| Jul 29, 2023 at 2:57 | comment | added | Ben C | A map of connected CW complexes induces a surjection on $\pi_1$ if and only if the pullback of any connected covering space is connected. Then use that if $f : X \to Y$ is a closed map of topological spaces with $Y$ connected and every fiber connected then $X$ is connected. | |
| Jul 29, 2023 at 2:11 | history | asked | Higgs-Boson | CC BY-SA 4.0 |