Timeline for Kunneth Theorem and localisation
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| May 7, 2017 at 7:03 | comment | added | user51223 | So, if we knew that $h(\theta_i)$ maps nontrivially under $H_*QS^0\to H_*L_EQS^0$ then it seems to me that we get a contradiction to the existence of $E$. But, I am not sure how much we can say about this. | |
| May 7, 2017 at 7:01 | comment | added | user51223 | It seemed to me that it might be possible to prove non-existence of such a spectrum if we use the destabilisation $\Omega^\infty$ and if we knew a couple of things. We know that (1) At the prime $2$, use $\theta_i\in{_2\pi_{2^{i+1}-2}^s}$ elements which we know are not decomposable for $i>3$, (2) The functors $\Omega^\infty$ and $L_E$ commute, (3) $\theta_i$ maps nontrivially under $h:{_2\pi_*^s}\to H_*Q_0S^0$ with $Q=\Omega^\infty\Sigma^\infty$, (4) the image of $h$ when restricted to decomposable elements is given by $h(\theta_i)$ for $i=1,2,3$. | |
| May 5, 2017 at 18:12 | comment | added | user51223 | Would perhaps the natural map fail to be monic in general, which then will prevent the map from being isomorphism?! | |
| May 5, 2017 at 14:32 | history | edited | Neil Strickland | CC BY-SA 3.0 |
Update to address the second question
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| May 5, 2017 at 11:26 | comment | added | user51223 | I corrected the first statement. The statement in Ravenel's book, on page 8 of the Orange Book, is that $H\mathbb{F}$ and $K(n)$ are ''essentially'' the only homology theories for which Kunneth theorem holds. I suppose the word ``essentially'' therein then needs more explanation, and your arguments almost provide a proof. | |
| May 5, 2017 at 10:29 | history | answered | Neil Strickland | CC BY-SA 3.0 |