Timeline for Is a manifold paracompact? Should it be?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jul 7 at 1:34 | comment | added | Dmitri Pavlov | @Z.M: Some examples of nontrivial bundles over nonparacompact spaces are given in arxiv.org/abs/1309.2524. Since every smooth function on the long line is eventually constant, I suspect the techniques of that article can be adapted to show that the higher cohomology of the structure sheaf of the long line is nonvanishing. | |
| Jul 6 at 15:41 | comment | added | Z. M | It seems that the crucial failure is affineness. I wonder whether the higher sheaf cohomology of the structure sheaf $C^\infty(-)$ is non-vanishing for the long line as well? | |
| Jul 6 at 14:18 | comment | added | Dmitri Pavlov | @Z.M: The tangent bundle of the long line is a counterexample to both statements: it does not admit a connection and its module of sections is not a finitely generated projective module. Indeed, if it did admit a connection, one could construct a metric by parallel transporting some metric on the fiber over some fixed point to all other points. If the module of sections of the tangent bundle of the long line was finitely generated and projective, we could induce a metric on it from its presentation as a direct summand of a trivial bundle, which does admit a metric. | |
| Jul 6 at 5:32 | comment | added | Z. M | Are the existence of connections and Serre–Swan provably false for some non-paracompact manifold? | |
| Nov 26, 2019 at 5:39 | vote | accept | AmorFati | ||
| Nov 25, 2019 at 20:53 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
added 126 characters in body
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| Nov 25, 2019 at 16:12 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |