Timeline for Variant of Leray-Hirsch for complex-oriented cohomology theory
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| May 6 at 21:55 | comment | added | Andres Mejia | Sorry if you were already aware of this proof. I don't know maybe someone smarter can cook up the weakest conditions for (something like) the second fact to hold and then this argument works | |
| May 6 at 21:50 | comment | added | Andres Mejia | I'm not sure about a reference, but the Atiyah-Hirzebruch spectral sequence for a fibration proves this theorem. You need to know that $E^q(P)$ surjects onto $H^0(B,E^q(F))$ which your assumption buys you. If you know that $E_2^{p,q}=H^P(B,E^q(F)) \cong H^p(B,E^*) \otimes_{E^*} H^q(F,E^*)=E_2^{p,0} \otimes E_2^{0,q}$ [ For example if the the cohomology of the fiber is free as an $E^*$ module, but maybe Achim's rmk is ok] then the spectral sequence collapsing on the second page ($d_2=0$ on $E^{p,0}$) proves the claim. I saw your last comment but this is the SS argument I know | |
| Apr 23 at 23:45 | comment | added | onefishtwofish | @AchimKrause I understand your point. Even in my restrictive case, to get $E^*(B)$ to be free over $E_*$, I need an assumption. I am happy to assume complex orientable as well and have adjusted the question. How should the argument go then? | |
| Apr 23 at 23:43 | history | edited | onefishtwofish | CC BY-SA 4.0 |
added 2 characters in body
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| Apr 23 at 18:40 | comment | added | Achim Krause | You definitely need some flatness assumption on the cohomology of $F$ or $B$. If $E$ is complex orientable, evenness of the cells of $B$ probably saves you there, but without assumptions on $E$ there should be counterexamples even with trivial fiber bundles $F\times B$ (for example with $E=\mathbb{S}$) | |
| Apr 23 at 18:18 | comment | added | onefishtwofish | @Tyrone I do not wish to assume the cohomology of the fiber is free over $\mathbb{E}_*$. | |
| Apr 23 at 18:07 | comment | added | Tyrone | It's proved on pg. 430 of tom Dieck's book that this is an isomorphism of left $E^*(B)$-modules (it is generally not an algebra isomorphism). | |
| Apr 23 at 17:04 | history | asked | onefishtwofish | CC BY-SA 4.0 |