Timeline for Restriction of a fibration to an open subset with connected diffeomorphic fibers
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 2, 2023 at 12:05 | comment | added | RKS | Yes, I see it now. | |
| Jul 2, 2023 at 11:57 | comment | added | R. van Dobben de Bruyn | You can almost do the same example over the reals as well. The curve $xy=1$ is the standard parabola $y = 1/x$. Away from $0$, this is a trivial fibre bundle $\mathbf R^\times \times \mathbf R^\times \to \mathbf R^\times$. Moreover, $H^1(f^{-1}(V),\mathbf Z) \neq 0$ for any small ball $V \subseteq \mathbf R$ around $0$. The reason it's not a counterexample is that $U$ is not connected in this case. | |
| Jul 2, 2023 at 11:53 | comment | added | RKS | Oh! It is that simple. I was thinking in the real plane sense and thought that the line $xy=1$ will delete some part of $f^{-1}(V)$, away from $(0,0)$. Thank you! | |
| Jul 2, 2023 at 11:33 | comment | added | R. van Dobben de Bruyn | The non-vanishing is just because the intersection of $f^{-1}(V)$ with a small ball around $(0,0)$ in $\mathbf C^2$ is diffeomorphic to $\mathbf R^4 \setminus \{0\}$. Away from $0$, the map $f^{-1}(\mathbf C^\times) \to \mathbf C^\times$ is in fact a trivial fibre bundle: the map $f^{-1}(\mathbf C^\times) \to \mathbf C^\times \times \mathbf C^\times$ given by $(x,y) \mapsto (x,xy-1)$ is a biholomorphism with inverse $(x,z) \mapsto (x,(z+1)/x)$. | |
| Jul 2, 2023 at 5:31 | vote | accept | RKS | ||
| Jul 2, 2023 at 5:30 | comment | added | RKS | Thanks a lot. Kindly share a reference for the cohomology non-vanishing result. Also I am curious to know if $0$ is the only troubling point. I mean if we did not add the point $(0,0)$, is the restriction a fibre bundle? | |
| Jul 1, 2023 at 23:11 | history | edited | R. van Dobben de Bruyn | CC BY-SA 4.0 |
Simplified proof.
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| Jul 1, 2023 at 21:52 | history | answered | R. van Dobben de Bruyn | CC BY-SA 4.0 |