Timeline for Is there a (satisfying) proof that cellular cohomology is isomorphic to simplicial cohomology that doesn't use relative cohomlogy?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 15, 2015 at 11:13 | comment | added | Achim Krause | The singular cochain complex isn't really a complex of sheaves, is it? What's the argument that sheafification doesn't break things if you want to show your way that sheaf cohomology = singular cohomology? | |
| May 15, 2015 at 9:40 | answer | added | Russ Woodroofe | timeline score: 1 | |
| May 15, 2015 at 3:04 | answer | added | Jesse C. McKeown | timeline score: 1 | |
| May 14, 2015 at 22:06 | answer | added | Ronnie Brown | timeline score: 0 | |
| May 14, 2015 at 1:40 | comment | added | few_reps | @NoamD.Elkies Indeed. And I had scruples sending this comment ... But, I said, if someone cares, I'll answer that $C_\infty$ is twice as interesting as $\mathbf Z$ ... | |
| May 14, 2015 at 1:30 | comment | added | Noam D. Elkies | @few_reps the infinite cyclic group is not nearly as interesting as the ring $\bf Z$. | |
| May 14, 2015 at 1:27 | answer | added | Peter May | timeline score: 15 | |
| May 14, 2015 at 1:00 | comment | added | Eric Wofsey | By the way, the proof in Hatcher is exactly the same as the spectral sequence proof Ben alludes to, just without using the word "spectral sequence". | |
| May 14, 2015 at 0:44 | comment | added | few_reps | @QiaochuYuan ... so many arithmeticians shot... | |
| May 13, 2015 at 23:02 | comment | added | Qiaochu Yuan | The Grothendieck group of finite CW complexes, suitably defined, is $\mathbb{Z}$. The isomorphism is given by taking the compactly supported Euler characteristic. It's not a very interesting group. | |
| May 13, 2015 at 22:13 | comment | added | Ben Webster♦ | Avoiding relative homology is a very bad idea by the way. The whole point is that the cellular chain groups are the homology of the $n$-skeleton rel the $n-1$-skeleton. | |
| May 13, 2015 at 22:10 | comment | added | Ben Webster♦ | There's a very short proof using the spectral sequence associated to the the filtration of the space by $n$-skeletons. The $\infty$-page is the singular homology, the $E_2$ page in the cellular homology, and its clear for degree reasons that the rest of the differentials are 0. Unfortunately there isn't time before going to bed (or room in this margin) to write all the details. I recall reading it in Fuks: amazon.com/Beginners-Course-Topology-Geometric-Universitext/dp/… | |
| May 13, 2015 at 22:05 | history | asked | Makhalan Duff | CC BY-SA 3.0 |