Timeline for mod $p$ homology of Thom spectra MSU
Current License: CC BY-SA 4.0
15 events
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May 16, 2020 at 1:47 | history | edited | user131113 | CC BY-SA 4.0 |
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May 4, 2020 at 22:20 | history | edited | user131113 | CC BY-SA 4.0 |
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Mar 13, 2020 at 19:05 | comment | added | user131113 | Let us continue this discussion in chat. | |
Mar 13, 2020 at 18:36 | comment | added | user43326 | Maybe I didn't get this right. The fact that it is the wedge of $BP$ gives the comodule structure in "coordinate free" way, or as abstract comodule over $A_p$. But maybe you are after "explicit" coaction with a chosen basis? | |
Mar 13, 2020 at 16:06 | comment | added | user131113 | user43326 all your hints are well known. Could you, please, write down explicit formulae? Probably you know that it is really difficult to describe the action of $\mathfrak A_p$ on the Chern classes, even using Wu's formula. Indeed it is easy to describe the action on the Chern roots, but after that you should rewrite the result in terms of elementary symmetric polynomials (Chern classes). Nevertheless, using the action on $\mathcal C P^\infty$ you can describe COaction on homology of $\mathcal C P^\infty$ and therefore of $MU$. | |
Mar 13, 2020 at 12:28 | comment | added | user43326 | Well, the point is that "Wu's formula" or whatever it is called, may make sometimes the action of the Steenrod algebra on homology of Thom spectrum easier to compute than that on the homology of the base space. | |
Mar 13, 2020 at 11:48 | comment | added | user131113 | @user43326 of course we need to compute homology of $MSU$ using the Thom isomorphism. But the Thom isomorphism just reduces to the case of homology of $BSU$, whose comodule structure is still not so easy to compute | |
Mar 13, 2020 at 7:10 | comment | added | user43326 | Well, Pengalley says, "standard method"... I would imagine you can simply compute it using the Thom isomorphism. Although there might be an argument using the fact that both $MSU$'s homotopy and homology are torsion-free. | |
Mar 12, 2020 at 15:22 | comment | added | user131113 | @user43326 do you how to prove the latter? | |
Mar 12, 2020 at 14:27 | comment | added | user43326 | I don't have any reference, but at odd $p$, $MSU$ splits as wedge of $BP$, this gives the action of the Steenrod algebra. | |
Mar 12, 2020 at 11:47 | comment | added | user131113 | @user43326, is there any reference to $A_p^*$-comodule structure of $H_*(MSU;\mathbb F_p)$ for odd $p$? | |
S Mar 12, 2020 at 11:45 | history | suggested | Igor Sikora | CC BY-SA 4.0 |
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Mar 12, 2020 at 11:22 | review | Suggested edits | |||
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Mar 12, 2020 at 8:02 | comment | added | user43326 | Basically you can play the same game at odd prime. I don't know about the explicit $F_p$ algebra structure, but $A_p$ commodule structure at odd prime is quite simple, and at 2, it is described in (1.5) of The Homotopy Type of MSU David J. Pengelley American Journal of Mathematics American Journal of Mathematics Vol. 104, No. 5 (Oct., 1982), pp. 1101-1123 | |
Mar 12, 2020 at 7:23 | history | asked | user131113 | CC BY-SA 4.0 |