Timeline for Is such a map null-homotopic?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 23, 2016 at 16:24 | vote | accept | Matt Hogancamp | ||
| Oct 29, 2014 at 15:34 | comment | added | Gabriel C. Drummond-Cole | If every skeleton of a CW-complex $X$ is contractible, then the map from $X$ to a point is a quasi-isomorphism and thus a homotopy equivalence by Whitehead's theorem so $X$ is contractible. | |
| Oct 29, 2014 at 14:59 | comment | added | B K | By duality, the situation is equivalent to the situation of two cochain-complexes, zero in negative degrees, and a cochain main which is for each $k\geq 0$ chain homotopic to a chain map which is zero in degrees $0,\ldots,k$. That in turn is just equivalent to $f$ being null-homotopic when restricted to the complex truncated at $k$-th level. Now suppose we had a CW-complex of infinite dimension whose cellular chain complex is not contractible but that of each $k$-skeleton is. Then we would have a counterexample. However, I don't know whether such CW-complexes exist. | |
| Oct 29, 2014 at 11:54 | answer | added | Jeremy Rickard | timeline score: 7 | |
| Oct 27, 2014 at 11:12 | comment | added | Fernando Muro | My feeling is that the answer is 'no' since thay would be a kind of 'phantom map'. I haven't been able to find an example, though, but I'd love to see one! | |
| Oct 25, 2014 at 15:19 | review | First posts | |||
| Oct 25, 2014 at 15:24 | |||||
| Oct 25, 2014 at 15:17 | history | asked | Matt Hogancamp | CC BY-SA 3.0 |