Timeline for Relation between the pushback closed form of sphere bundle and the pullback closed form of ball bundle
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| Jun 14, 2022 at 12:59 | comment | added | Aleksandar Milivojević | Maybe there are some additional assumptions you are implicitly making to have that map in the Gysin sequence be an isomorphism. Otherwise take for example the unit disk bundle of the tangent bundle of $S^2$; the boundary $\partial N$ is $RP^3$, and $H^2(RP^3)$ is not trivial while $H^1(S^2)$ is. Or, something that persists to real cohomology, $H^1(RP^3) = 0$ while $H^0(S^2) = Z$. Then one can cross everything with a circle to get different $H^2(\partial N)$ and $H^1(B)$. | |
| Jun 13, 2022 at 10:24 | comment | added | DLIN | @MarkGrant The volume form is tangent to the fiber, we can construct it by using a fiber-wise metric. Recall tha Gysin sequence gives the isomrophism $H^{n+m-1}(\partial N)\cong H^n(B)$. | |
| Jun 13, 2022 at 8:19 | comment | added | Mark Grant | Ok, thanks. But now I realise I don't understand the definition of $\omega_b$. Inside the brackets you take the wedge of a form on $\partial N$ with a form on $S^{m-1}$. How does one do this if the sphere bundle is non-trivial? | |
| Jun 13, 2022 at 7:46 | history | edited | DLIN | CC BY-SA 4.0 |
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| Jun 13, 2022 at 6:12 | history | edited | DLIN |
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| Jun 13, 2022 at 4:38 | history | edited | DLIN | CC BY-SA 4.0 |
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| Jun 13, 2022 at 2:24 | comment | added | DLIN | Sorry, since have a section $s:B\to N$, the second one means $\int_B(s^*\omega)$, and as $\omega$ is closed, we can show that the integral is indepedent of the section. @MarkGrant | |
| Jun 12, 2022 at 18:28 | comment | added | Mark Grant | Should the second integral in the Question be over $N$, rather than $B$? | |
| Jun 12, 2022 at 15:22 | history | edited | DLIN | CC BY-SA 4.0 |
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| Jun 12, 2022 at 14:16 | history | edited | DLIN | CC BY-SA 4.0 |
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| Jun 12, 2022 at 10:42 | history | edited | DLIN |
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| Jun 12, 2022 at 9:52 | history | edited | DLIN |
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| Jun 12, 2022 at 5:58 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
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| Jun 12, 2022 at 3:38 | history | edited | DLIN | CC BY-SA 4.0 |
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| Jun 12, 2022 at 2:52 | history | edited | DLIN | CC BY-SA 4.0 |
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| Jun 12, 2022 at 2:39 | history | asked | DLIN | CC BY-SA 4.0 |