Timeline for A homology theory which satisfies Milnor's additivity axiom but not the direct limit axiom?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 1, 2016 at 12:43 | history | edited | Karol Szumiło | CC BY-SA 3.0 |
deleted 100 characters in body
|
| Jan 29, 2016 at 16:00 | history | edited | Karol Szumiło | CC BY-SA 3.0 |
deleted 3 characters in body
|
| Jan 29, 2016 at 15:53 | comment | added | Karol Szumiło | Perhaps I should add that my answer is really about non-reduced homology and I'm secretly using relative homology all the time, but then the conclusion translates to reduced homology theories as usual. | |
| Jan 29, 2016 at 15:49 | history | edited | Karol Szumiło | CC BY-SA 3.0 |
added 4 characters in body
|
| Jan 29, 2016 at 15:42 | history | edited | Karol Szumiło | CC BY-SA 3.0 |
added 2315 characters in body
|
| Jan 29, 2016 at 15:19 | comment | added | Bruno Stonek | Actually, now that I look more closely, if we look at Switzer, Remark 1 p. 331, he says: "One can show that (the wedge axiom) implies (the direct limit axiom): the proof is a generalization of (the fact that homology commutes with colimits over a countable tower) using a "generalized telescope"", so, a homotopy colimit. I hadn't seen this before... | |
| Jan 29, 2016 at 15:05 | comment | added | Karol Szumiło | Also, my answer is not "model-category-heavy". You do not need to know anything about model categories to make sense of this. I will try to elaborate to make this clear. | |
| Jan 29, 2016 at 15:04 | comment | added | Karol Szumiło | Well, in the times of Adams things like that were not sorted out yet. Why Hatcher says that this axiom is stronger (or if he actually claims that it is strictly stronger), I cannot say. But in the recent years, abstract homotopy theory (including Goodwillie calculus which puts homology theories in an abstract context) progressed to a point that things like that became folklore. | |
| Jan 29, 2016 at 14:47 | comment | added | Bruno Stonek | Well, that part I understand. I'm still puzzled by the fact that additivity would imply taking all filtered homotopy colimits to colimits. Why would Adams have introduced the axiom in his 1970 paper then? Also, more recent books like Hatcher also insist on the necessity of this "stronger" form (p. 455). (What I don't understand is the more model-category-heavy part. But there's too much knowledge I still lack in that department, so don't worry trying to expand on it too much. Thanks, though.) | |
| Jan 29, 2016 at 14:40 | comment | added | Karol Szumiło | Yes, my statement is rather abstract, but any CW-complex is the homotopy colimit of its finite subcomplexes so the "direct limit axiom" follows. If you have any specific questions, I can try to elaborate. | |
| Jan 29, 2016 at 14:36 | comment | added | Bruno Stonek | I don't have enough familiarity with some of the objects you use in your answer to understand it just yet. But if I understand correctly, you're actually saying that Milnor's additivity axiom is equivalent to the direct limit axiom, correct? | |
| Jan 29, 2016 at 13:17 | history | answered | Karol Szumiło | CC BY-SA 3.0 |