Timeline for A manifold whose tangent space is a sum of line bundles and higher rank vector bundles
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jul 22 at 21:16 | comment | added | Ali Taghavi | Thank you very much for your answer | |
| Jul 22 at 17:43 | comment | added | Robert Bryant | @AliTaghavi: A simple example, is to let $X = S^3$ and $B=S^2$ and let $\pi:S^3\to S^2$ be the Hopf map. The tangent bundle to $S^2$ is irreducible over $S^2$, but the pullback via $\pi$ is trivial over $S^3$. | |
| Jul 22 at 15:55 | comment | added | Ali Taghavi | In this example I think the plane $A$ is the pull back of the real tangent bundle of $\mathbb{C}P^2$ under the fiber map $S^5 \to \mathbb{C}P^2$. The tangent bundle of CP^2 is irreducible but it apparently does not implies that the bundle A you mentioned is irreducible. So this is a motivation to ask: What is an example of principle bundle $X\to B$ where $A$ is a connection plane tangent to X such that $TB$ is irreducible but A is not? I would appreciate if you give comment on this question | |
| Jul 21 at 9:16 | comment | added | Ali Taghavi | I just ask this question and its extension here mathoverflow.net/questions/475457/… | |
| Jul 21 at 7:17 | comment | added | Ali Taghavi | However the tangent bundle of $S^5$ is not decompossible in the way you mentioned but according to the Splitting principal there is an space $X$ and continuous map $f:X\to S^5$ such that $f^*$ is an isomorphism in K theory and the pull back $f^*(TS^5)$ of the tangent bundle of $S^5$ is decomposed to direct sum of line bundle. In this case s there an explicit (compact) manifold $X$ with this property?What is theminimum possible dimension for such manifold $X$? By this question I search for a possible smooth version of splitting principle in K theory | |
| Jul 7 at 20:02 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Removed a spurious sentence at the end. (Editing error.)
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| Jul 7 at 16:46 | history | answered | Robert Bryant | CC BY-SA 4.0 |