Timeline for Which stable homotopy groups are represented by parallelizable manifolds?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 4, 2020 at 13:36 | comment | added | Connor Malin | Ah, after reading @archipelago's answer I realize that since tangent bundles are the same rank as the dimension of the manifold, you are stably trivial if and only if adding a single trivial bundle makes you trivial. | |
| Jul 4, 2020 at 13:17 | comment | added | Connor Malin | Forgive me if I am just stating the obvious here, but wouldn't the GMTW-theorem for cobordisms with honest framings of the tangent bundle immediately show that all are represented by manifolds which have tangent bundle trivial after adding a single trivial bundle. Obviously not as strong as the result you prove, but at least a sanity check. | |
| Jun 27, 2020 at 17:36 | vote | accept | Chris Schommer-Pries | ||
| Jun 26, 2020 at 20:52 | comment | added | Panagiotis Konstantis | It seems to me that the obstruction in odd dimension could be the semi-characteristic of the manifold. Bredon and Kosinski showed in (jstor.org/stable/1970531) that a stably framed manifold of dimension $n$ admits $n$ linear independent vector fields if only if the ($\mathbb Z_2$-valued) semi-characteristic vanishes. | |
| Jun 26, 2020 at 20:20 | history | answered | Oscar Randal-Williams | CC BY-SA 4.0 |