Timeline for The word "torsion" and its connection to geometry and homology
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 8, 2018 at 18:45 | comment | added | Watson | Possibly related: math.stackexchange.com/questions/300586 | |
| May 25, 2010 at 14:10 | answer | added | KConrad | timeline score: 6 | |
| May 25, 2010 at 13:05 | answer | added | Charles Matthews | timeline score: 6 | |
| May 25, 2010 at 12:36 | comment | added | BCnrd | [harmless typo: In the above I should have said "paracompact Hausdorff topological space or manifold..." so that one has partitions of unity so as to kill high cohomology of $O_X$.] | |
| May 25, 2010 at 12:34 | comment | added | BCnrd | First, the "twistedness" of the Mobius strip $M$ is encoded in how it fibers over $S^1$ as nontrivial topological (or smooth) line bundle. Second, the exponential sequence $0 \rightarrow O_X \rightarrow O_X^{\times} \rightarrow \mathbf{Z}/2\mathbf{Z} \rightarrow 0$ for any topological space or smooth manifold $X$ (with $O_X$ the sheaf of continuous or smooth functions) yields an isomorphism ${\rm{Pic}}(X) = {\rm{H}}^1(X,O_X^{\times}) \rightarrow {\rm{H}}^1(X,\mathbf{Z}/2\mathbf{Z})$. Thus, the line bundle $M$ represents the nontrivial class in ${\rm{H}}^1(S^1,\mathbf{Z}/2\mathbf{Z})$. | |
| May 25, 2010 at 12:05 | history | asked | Akela | CC BY-SA 2.5 |