Timeline for Splittings in the difference bundle construction of Atiyah-Hirzebruch
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jan 25, 2019 at 19:25 | history | edited | Neil Strickland | CC BY-SA 4.0 |
Corrected spelling of Hirzebruch
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| Jan 25, 2019 at 16:40 | comment | added | keaton | I think I understood what I missed. I appreciate for all comments :) | |
| Jan 25, 2019 at 16:21 | comment | added | Tobias Shin | It is true that every short exact sequence of complex vector bundles split, by the above argument. However, it is not true that every short exact sequence of holomorphic vector bundles split, as in algebraic geometry. | |
| Jan 25, 2019 at 16:15 | comment | added | keaton | I forgot to write something. In this case, $E_i$'s are complex vector bundles. Is it possible to induce complex structure to the orthogonal complement of $E_i$? It looks little strange for me because then, if the base space is paracompact, then every short exact sequence splits!. In general, it never happens in algebraic geometry. What is a difference when there is a metric? | |
| Jan 25, 2019 at 15:33 | history | edited | keaton | CC BY-SA 4.0 |
deleted 1 character in body
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| Jan 25, 2019 at 13:16 | comment | added | Panagiotis Konstantis | If your base space is paracompact then it is possible to equip $E_i$ with a metric. Consider the decomposition of $E_i$ into $F_i$ and its orthogonal complement which is isomorphic to $F_{i-1}$. Now it is possible to construct a splitting map. | |
| Jan 25, 2019 at 12:31 | history | asked | keaton | CC BY-SA 4.0 |