Timeline for Extending topological vector bundles and obstruction theory
Current License: CC BY-SA 4.0
4 events
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| Jan 18 at 1:55 | comment | added | Connor Malin | Yes, the theorem provides complete obstructions relevant to extending maps up to a homotopy which stays constant on a subspace. You might find it helpful to think about this theorem in the context of mapping into the Eilenberg-MacLane space $K(A,n)$ which is perhaps a more familiar topic. | |
| Jan 18 at 1:48 | comment | added | Paul Cusson | @ConnorMalin Thank you for the reference! And that's good news to hear about how there's not more than needed than this theorem. I sometimes hear the words "complete obstruction" or something similar in reference to an obstruction being necessary and sufficient for its intended purpose. I was lead to believe this means having this obstruction prevents the object in question from existing, and failing to have this obstruction always allows the object to exist. Is the correct definition? And can one obtain complete obstructions for the problem in my question with what you mentioned? | |
| Jan 18 at 1:41 | comment | added | Connor Malin | It seems to me like the fundamental theorem of obstruction theory, say Theorem 7.1 of Kirk and Davis, is very literally the exact situation of your question when you remember that vector bundles are classified by maps to $BO(n)$. I think there is not much more to say. We know a fair few homotopy groups of $BO(n)$, especially when $n=\infty$ so it often turns out that one can actually work with the obstructions in practice. | |
| Jan 16 at 20:57 | history | asked | Paul Cusson | CC BY-SA 4.0 |