Timeline for The higher Van Kampen Theorems and computation of the unstable homotopy groups of spheres
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 8, 2020 at 17:19 | comment | added | Ronnie Brown | This question really goes back to the 1932 ICM at Zurich when E Cech gave a seminar on higher homotopy groups.,. Their abelian nature suggested that they were not a suitable generalisation to higher dimensions of the fundamental group. We now know that seems to require higher homotopy groupoids of structured spaces. As homotopy groups gradually became an important subject, the other idea became forgotten! | |
| Sep 27, 2010 at 20:04 | comment | added | Ronnie Brown |
Carl writes cat-n groups can only provide information on the n-type'. I wonder about the word only'! The HHSvKT computes (when it works) the n-type of a colimit from the n-types of the pieces and the gluing information. Sometimes (this has been done for cat$^2$-groups) this allows a lot of information on homotopy groups of the colimit, Whitehead products, composition operators, ... Is not that amazing? Also not many have worked on these gadgets.
|
|
| Sep 27, 2010 at 9:42 | comment | added | Ronnie Brown | I feel this is a question of horses for courses. If you wanted to compute pi_200 of the 2-sphere this way you would need a 200-cube structure on the 2-sphere, e.g. from 200 subspaces, and it is not so clear how to get this. However SX does have a convenient triad structure, with two cones, and this is used in classical homotopy theory. The traditional SvKT for the fundamental group does not compute the fundamental group of the circle. The HHSvKT computes some things not otherwise computable, e.g. some nonabelian n-ad groups. Better to look at what these do rather than don't do! | |
| Sep 24, 2010 at 18:12 | comment | added | Tim Porter | @Carl. Hi. No you are exactly right, but pessimistically so. :-) (See my reply to Tom above) The theory has not been developed that far as to be able to see the ways of extracting information from the gadget. There is probably too much data in the model and one has to see how to use algebraic methods to prise out of it information that is needed. That is not that unusual in algebra. With some of the gigantic simple groups there is a lot of energy that goes into, for instance, understanding what subgroups they have and that sort of effort leads to new tools for use elsewhere. | |
| Sep 24, 2010 at 16:05 | history | answered | Carl Futia | CC BY-SA 2.5 |