Suppse E is a cohomology theory which has Kunneth Formula, i.e $ E(A \wedge B)= E(A) \otimes_{E(pt)} E(B) $. What happens to the Atiyah Hirzebruch Spectral sequence while we compute $ E(A \wedge B) $? The
$ E^2 $ page of the spectral sequence has the relatively simple form in terms of the singular cohomology theory of $H^{*}(A \wedge B, \pi_{*}(E))$ which in turn can be expressed in terms of $H^{*}A, H^{*}B$.
Does the spectral sequence splits? $ \\ \\ $
It is been discussed in Tensor product of spectral sequences? that, there is no natural construction for the tensor product of spectral sequence.
In my special situation can we define it for Atiyah Hirzebruch Spectral sequence?
I am interested in the case when A,B are finite CW spectra. Kindly refer me the source where I can get some idea.
$\begingroup$
$\endgroup$
3
-
$\begingroup$ Actually the thread you quote says that there is natural construction for the tensor product in the case you are talking about. $\endgroup$– user43326Commented Sep 4, 2019 at 13:18
-
$\begingroup$ It may not be a spectral sequence because of Kunneth formula. $\endgroup$– Monkey.D.LuffyCommented Sep 4, 2019 at 16:41
-
$\begingroup$ The point is that since A and B are supposed to be finite, the Kunneth isomorphism holds only in very few cases (there won't be situations like Landweber-flat), so you get Kunneth isomorphism everywhere in sight. That is either $\pi _*(E)$ is field, or one of $H_^*(A)$ or $H^*(B)$ is torsion-free. Thus $H^*(A\times B \pi_*(E))\cong H^*(A),\pi _*(E))\times H^*(B),\pi _*(E))$ $\endgroup$– user43326Commented Sep 4, 2019 at 18:48
Add a comment
|