Timeline for How do you intepret "kill a cohomology class" intuitively for attaching an n-cell?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 7, 2012 at 17:57 | vote | accept | Xuxu | ||
| Oct 7, 2012 at 13:01 | answer | added | Liviu Nicolaescu | timeline score: 0 | |
| Oct 6, 2012 at 12:49 | answer | added | Andreas Blass | timeline score: 9 | |
| Oct 6, 2012 at 10:56 | answer | added | CuriousUser | timeline score: 4 | |
| Oct 6, 2012 at 7:00 | history | edited | Mark Grant |
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| Oct 6, 2012 at 3:51 | comment | added | Will Sawin | Formally, it kills a class if it's the image under the coboundary map of that class, and it creates a new class if it's not the image under the coboundary map of any class. So a picture of whether it kills or attaches a class is a picture of the coboundary map. So in CW theory, whether it kills a class depends on whether the attaching map is homologically trivial or not. In Morse theory, it depends on the flow lines coming out of that critical point and whether they cancel each other. | |
| Oct 6, 2012 at 3:47 | comment | added | Will Sawin | $H^*$ being $\mathbb Q$ implies that the map is either injective or $0$, since maps from $\mathbb Q$ to $\mathbb Q$-vector spaces are injective. Doesn't $H^*$ being $\mathbb Q$ follow from the fact that $\mathcal M^{+}$ minus $\mathcal M^{-}$ contains just a single critical point, in Morse theory? Or a single cell, in CW complex cohomology? What cohomology theory are you using here? | |
| Oct 6, 2012 at 3:10 | history | asked | Xuxu | CC BY-SA 3.0 |