Timeline for Spaces with free MU-homology
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 15, 2012 at 8:53 | comment | added | Tilman | $E$=mod-2 K-theory and $X = RP^\infty$? | |
| Jan 13, 2012 at 22:30 | comment | added | Lennart Meier | Ah, I see: because every homology theory is rationally ordinary. | |
| Jan 13, 2012 at 19:01 | comment | added | Dan Ramras | Lennart, there is a very general and precise theorem of Arlettaz that gives bounds on the "order" of the differentials in any AHSS. Here order means the smallest number R so that Rd = 0, where d is the differential. The original, less precise version (which suffices for Neil's statement) goes back to Dold, according to the introduction of Arlettaz's paper. Here's the reference: Arlettaz, Dominique, The order of the differentials in the Atiyah-Hirzebruch spectral sequence. K-Theory 6 (1992), no. 4, 347–361. | |
| Jan 13, 2012 at 14:33 | comment | added | Lennart Meier | Thanks for your comment! Can you tell me why the differentials in the AHSS are torsion-valued? | |
| Jan 13, 2012 at 12:43 | comment | added | Neil Strickland | Note that Atiyah-Hirzebruch differentials are always torsion-valued, so if $H_*(X)$ is free abelian then the AHSS for $MU_*(X)$ will collapse showing that $MU_*(X)$ is free over $MU_*$, and it follow that $E_*(X)$ is free over $E_*$ on the same basis for any $MU$-algebra $E$ (even if $E_*$ is torsion). For example, $MU_*(U(n))$ is free and not in even degrees. However, a theorem of Miller gives a stable splitting of $U(n)$ into Thom spectra each of which is either in even degrees or odd degrees, so we don't exactly get an answer to your question. Other Lie groups might do the trick. | |
| Jan 13, 2012 at 10:57 | history | asked | Lennart Meier | CC BY-SA 3.0 |