Timeline for Homotopy equivalence from contractibility of fiber
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 4, 2013 at 4:01 | comment | added | Andy Putman | Properness is certainly needed; it is not hard to construct counterexamples without it. I do not know whether assuming that $f$ is cellular is good enough. | |
| Apr 4, 2013 at 3:47 | comment | added | Cusp | @Andy: Thanks. I am a little curious about the property that the map $f$ should proper. How important this condition really is? What I mean is can I replace this condition by a more simple condition like taking $f$ to be cellular or something of that sort. | |
| Apr 4, 2013 at 3:42 | vote | accept | Cusp | ||
| Apr 4, 2013 at 3:42 | vote | accept | Cusp | ||
| Apr 4, 2013 at 3:42 | |||||
| Apr 4, 2013 at 3:31 | comment | added | Andy Putman | @Vel Nias : It depends on your conventions. All right-thinking people consider the empty set non-connected (and certainly not contractible), but just in case the OP has other ideas I thought I'd avoid this issue by assuming surjectivity. | |
| Apr 4, 2013 at 3:02 | comment | added | Vidit Nanda | Andy: this is how I have seen (and stated) this theorem also, but does one really need to say "surjective"? The empty set does not have the homotopy type of a point, does it? | |
| Apr 3, 2013 at 22:13 | comment | added | Ricardo Andrade | I just want to add a small, tangential refinement. In Andy's conclusion for finite dimensional complexes, we can actually do a bit better. If the fibres of a map between finite CW-complexes are all contractible and locally contractible, then the map is actually a "cell-like map", and thus a simple homotopy equivalence. | |
| Apr 3, 2013 at 21:10 | history | answered | Andy Putman | CC BY-SA 3.0 |