Timeline for Chromatic orientability of manifolds
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 20, 2021 at 1:03 | comment | added | Ben Wieland | 1. If a smooth manifold is $E(1)$-orientable, is it $K(n)$ orientable for all $n>1$, since the structure group is the image of $J$? You need to account for the difference between additive and multiplicative structure, but that's not much, is it? Is that just regular orientability? . . . 2. For a simply connected manifold, is orientability the same as equivalence between $M\otimes E$ and $Hom(M,E)$? The latter seems well-suited to combining $K(n)$ into $BP$. But your manifolds are not simply connected. | |
| Jul 17, 2021 at 22:50 | history | edited | Nicholas Kuhn | CC BY-SA 4.0 |
added 301 characters in body
|
| Jul 17, 2021 at 17:08 | comment | added | Jeremy Hahn | Can you get orientations with respect to the Morava E-theories? Maybe Corollary 3.4 here is useful arxiv.org/pdf/1509.05678.pdf, which says that p-completed BP is a retract of a product of Morava E-theories. | |
| Jul 16, 2021 at 21:03 | comment | added | Ben Wieland | Your particular example is a complex manifold, indeed a hermitian symmetric space. The orientation double cover (where the parity enters) is $SO(m)/SO(2)\times SO(m-2)$. The stabilizer of a point contains $SO(2)$, which induces a complex structure on the tangent space. Since it is central in the stabilizer, it commutes with the isotropy and thus is canonical. | |
| Jul 15, 2021 at 22:02 | answer | added | Oscar Randal-Williams | timeline score: 7 | |
| Jul 15, 2021 at 19:16 | comment | added | Ryan Budney | That's exactly the sort of question I would pose to Nick Kuhn, if I had thought of it. | |
| Jul 15, 2021 at 18:58 | history | edited | Nicholas Kuhn | CC BY-SA 4.0 |
added 142 characters in body
|
| Jul 15, 2021 at 18:18 | history | asked | Nicholas Kuhn | CC BY-SA 4.0 |