Timeline for How do people deal with different cohomology theories [closed]
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Nov 28, 2013 at 20:06 | comment | added | Al-Amrani | Sure that you cannot easily transfer computation methods from a cohomology theory to another one. In the topological case, K-theory (a generalized cohomology theory) is more difficult to compute than integral cohomology, although the first is easier to define (construct) than the second ! | |
| Nov 28, 2013 at 15:06 | review | Reopen votes | |||
| Nov 29, 2013 at 1:27 | |||||
| Nov 28, 2013 at 6:03 | comment | added | Greg Friedman | I think it's unfair to point the poster to the Eilenberg-Steenrod axioms without pointing out that compactly supported cohomology (which s/he asks about specifically) does not satisfy those axioms and so is not a "cohomology theory" in that sense. | |
| Nov 27, 2013 at 20:54 | comment | added | GFR | I thought that what I am asking -given that cohomology is not my field but a tool and I need to employ it as a relatively basic level, is it possible to transfer basic results from one cohomology theory to the other or does one need heavy machinery to do that or are only higher level results which do transfer- is a question that often arises in research: you need a tool which is not in your main area, is it worth learning it? In my case the tool is basic use of different cohomology theories. However I am fine with this being moved if you think it is not appropriate. | |
| Nov 27, 2013 at 20:37 | history | closed |
Ricardo Andrade Fernando Muro David White Andy Putman Will Jagy |
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| Nov 27, 2013 at 20:20 | comment | added | Al-Amrani | It depends on the specific problem you are interested in.For example, in algebraic geometry, when the ground field is not necessarily C (complex numbers), you need étale cohomology (constructed by GROTHENDIECK via étale topology) (instead of the ordidary complex one you must first learn about), to solve , let us say, Weil's conjectures.It is a theorem that over C the two cohomlogies are isomorphic ! | |
| Nov 27, 2013 at 20:10 | review | Close votes | |||
| Nov 27, 2013 at 20:40 | |||||
| Nov 27, 2013 at 20:03 | comment | added | GFR | Thanks Ben, I actually knew that cohomology can be axiomatised, part of my question was indeed to understand wether the common structure is large enough that learning to calculate in one makes it very easy learning how to calculate in the other or not, sorry if that was not clear. I am more interested in using cohomology as a tool rather then in its more formal aspects. | |
| Nov 27, 2013 at 19:58 | comment | added | Ben McKay | en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod_axioms | |
| Nov 27, 2013 at 19:43 | history | asked | GFR | CC BY-SA 3.0 |