Timeline for De Rham homology
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 28, 2010 at 9:45 | vote | accept | Dmitri Pavlov | ||
| Feb 28, 2010 at 9:45 | comment | added | Dmitri Pavlov | Oops, I accidentally reversed the directions of arrows. With this correction everything becomes trivial. | |
| Feb 28, 2010 at 9:12 | comment | added | Leonid Positselski | No, the differential will be raising the degrees of the forms twisted with orientations, not lowering them. What you get in this way is simply the de Rham complex with coefficients in the orientations (notice that the bundle of orientations has a natural flat connection). So your `de Rham homology' are simply the cohomology of the orientation local system (which are of course isomorphic to the Borel-Moore homology with constant coefficients, by Poincare duality). | |
| Feb 28, 2010 at 8:53 | comment | added | Dmitri Pavlov | I never claimed that I get polyvector fields. My statement was about polyvector fields twisted by D. Certainly, applying Hodge duality we can map this bundle isomorphically to the bundle of differential forms of complementary degree, but they still have to be twisted by the orientation bundle, so that the complex now looks like this: 0→Ω^n(M)⊗W→Ω^{n-1}(M)⊗W→⋯→Ω^0(M)⊗W→0, where W is the orientation bundle. Perhaps this is nothing else but the Poincaré duality expressed in this language. Either way of writing down this complex is fine with me. | |
| Feb 28, 2010 at 8:37 | history | answered | Leonid Positselski | CC BY-SA 2.5 |