Timeline for Reference: Relative cohomology of a morphism
Current License: CC BY-SA 3.0
9 events
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| Mar 13, 2016 at 13:25 | comment | added | Tintin | Thank you both Denis and Fernando. Still I would like to see a reference.For example, the relative cohomology at Hatcher's is defined for a closed subspace. Hatcher seems not to make the remark you mention, Fernando. Note that cohomology with compact support is a particular case of the relative cohomology of a morphism (the morphism being the closed embedding of the open complement).You can see at Hatcher's page 242 he defines cohomology with compact support and gives a more elaborated description. He doesn't even seem to write the associated long exact sequence. | |
| Mar 13, 2016 at 11:25 | comment | added | Fernando Muro | As Denis says, relative (co)homology is widely used when talking of pairs given by a space and a(n appropriate) subspace, but it is commonly remarked in most books that if you have a map, you can do de same by taking the mapping cylinder, which turns it into an inclusion. Any book will do, e.g. Hatcher, Spanier, Switzer, Whitehead... | |
| Mar 12, 2016 at 22:06 | comment | added | Denis Nardin | My guess is that everyone would understand what you mean. Historically the term relative cohomology was used only when the morphism is some kind of embedding. However up to homotopy all morphisms are embeddings (just embed in the mapping cylinder!) so nowadays most people don't really make the distinction. | |
| Mar 12, 2016 at 20:45 | comment | added | Tintin | Thank you once again, Denis. Yes, Adams puts just above Prop III.3.10 the cofiber sequence. However, he doesn't seem to call it there relative cohomology . The question is about the notation relative cohomology : is it standard in classic cohomology? In other words, if I speak with a classical topologist, can I use the term relative cohomology of a morphism? or should I specify first what do I mean and why do I call it that way? | |
| Mar 12, 2016 at 20:33 | history | edited | Tintin | CC BY-SA 3.0 |
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| Mar 12, 2016 at 20:25 | comment | added | Denis Nardin | Uh, again in the homotopy theory setting this is just the long exact sequence associated to a cofiber sequence. See for example propositions 3.9 and 3.10 in Adams' book, but really this is treated in any homotopy theory textbook. I don't know if there's an easy way to generalize it to étale cohomology or deRham cohomology. | |
| Mar 12, 2016 at 20:20 | comment | added | Tintin | Thank you Denis, but that I already know. I took a look at Adam's "Stable homotopy..." and he doesn't seem to speak about he subject, does he? Do you know a reference where they treat this? | |
| Mar 12, 2016 at 20:15 | comment | added | Denis Nardin | In the classical (homotopy theory) case this is the same thing as the cohomology of the mapping cone of $f$. This works also in equivariant and motivic homotopy theory. | |
| Mar 12, 2016 at 20:11 | history | asked | Tintin | CC BY-SA 3.0 |