Timeline for Poincaré duality with boundary conditions
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Dec 30, 2015 at 10:43 | history | suggested | John B | CC BY-SA 3.0 |
minor adjustments
|
| Dec 30, 2015 at 10:05 | review | Suggested edits | |||
| S Dec 30, 2015 at 10:43 | |||||
| Feb 26, 2011 at 20:40 | vote | accept | Pavol S. | ||
| Feb 22, 2011 at 14:12 | comment | added | Pavol S. | If $V_F=V$ for all $F$'s we still get de Rham cohomology tensored with $V$, and de Rham with compact support in the interior tensored with $V^*$, so the duality holds. The motivation for the question was from symplectic form on the moduli space of flat connections on a surface, with various boundary conditions, and Lagrangian subspaces coming from cobordisms. | |
| Feb 20, 2011 at 0:14 | answer | added | Johannes Nordström | timeline score: 6 | |
| Feb 18, 2011 at 3:44 | comment | added | John Klein | I guess I don't really have a feel for your question. What is the relation between $V$ and the geometry of the stratification of the manifold? What if $V_F = V$ for all faces $F$? How did this question arise? | |
| Feb 14, 2011 at 21:58 | comment | added | Pavol S. | @John Klein: Then $V$ is not important. One cohomology is de Rham's tensored with $V^*$ and the other one is (isomorphic to) de Rham's with compact support in the interior of $M$ tensored with $V$. We can safely put $V=\mathbb R$ in this case. More-dimensional $V$ is only needed for those more complex boundary conditions. For $V=\mathbb{R}$ we can only have $V_F=0$ or $V_F=\mathbb{R}$. In that case, if say $\partial M$ is divided by a hypersurface to two faces, one with $V_F=0$ and the other with $V_F=\mathbb{R}$, then we do get perfect pairing (at least I hope :) | |
| Feb 14, 2011 at 21:16 | comment | added | John Klein | You write: "If $V_F = 0$ for all $F's$ we get the standard Poincare duality for manifolds with boundary." But what is the role of $V$ in that statement? | |
| Feb 14, 2011 at 15:14 | history | edited | Pavol S. | CC BY-SA 2.5 |
added 12 characters in body
|
| Feb 13, 2011 at 20:52 | history | edited | Pavol S. | CC BY-SA 2.5 |
added 6 characters in body; edited tags
|
| Feb 13, 2011 at 16:02 | history | edited | Pavol S. | CC BY-SA 2.5 |
added 328 characters in body; added 28 characters in body
|
| Feb 12, 2011 at 23:55 | history | edited | John Klein |
edited tags
|
|
| Feb 12, 2011 at 15:10 | history | edited | Pavol S. | CC BY-SA 2.5 |
added 584 characters in body; deleted 4 characters in body
|
| Feb 11, 2011 at 21:04 | history | asked | Pavol S. | CC BY-SA 2.5 |