Timeline for computing homotopy type
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
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| Mar 8, 2011 at 15:54 | comment | added | Ryan Budney | If the target space is a sphere and the domain space a finite CW-complex then this is a totally reasonable problem. If the target space is a circle then this reduces to a computation in 1-dimensional cohomology, which is an algorithmic problem. If the target space is a higher-dimensional sphere then it's also a managable problem but could potentially require a fair bit of work. | |
| Mar 8, 2011 at 12:54 | comment | added | Peter Franek | Well, what you are saying (if I got it) is, that for a particular map from a circle to a topological space, you can not algorithmically determine, whether it is trivial (if the map is given by a finite set of data, such as simplicial map between simplicial complexes or so). Do you think that if the "second" space is not arbitrary, but a sphere and the "first space" is not a circle but a more arbitrary space, it does not change/simplify the situation? Thanks a lot for all the answers. | |
| Feb 26, 2011 at 17:50 | comment | added | Ryan Budney | You're being extremely imprecise, Peter. Presumably you do not mean "space of maps" but perhaps "homotopy-classes of maps"? If you want any object of this form to be a group you have to specify a multiplication -- what kind of multiplication do you want to use? I'm not aware of any general group structure on these kinds of objects. see: en.wikipedia.org/wiki/Cohomotopy_group | |
| Feb 22, 2011 at 11:31 | comment | added | Peter Franek | Sorry for the term "equivalent", I mean "homotopic". Is the space of homotopy classes from a manifold to a sphere a group at all? How would an algorithm that determines, whether a map from a manifold to sphere (not vice versa) i trivial, imply the "word problem" solution? Thanks, Peter | |
| Feb 11, 2011 at 16:18 | comment | added | Ryan Budney | Traditionally the word "homotopy equivalent" does not apply to paths. If you mean "homotopic" then you'd need the sphere to have dimension $n \geq 2$ for it to be true. You could allow $n=1$ if your homotopy does not fix endpoints but from your original question you appear to rule that out. Perhaps this is a terminology issue -- increasingly you seem to be using non-standard terminology. | |
| Feb 11, 2011 at 14:13 | comment | added | Peter Franek | Well, any two pathes in a shpere are homotopy equivalent. Does the space of "homotopy classes" from a manifold to the sphere form a "group" at all? Maybe I'm missing something important.. | |
| Feb 11, 2011 at 4:49 | comment | added | Ryan Budney | If you're interested in knowing whether or not two paths are homotopic, yes, you're dealing with the word problem and algorithmically there are fundamental problems (unless you know you're dealing with particularly special cell complexes but from your description it sounds like you're dealing with completely general complexes). | |
| Feb 11, 2011 at 0:46 | comment | added | Peter Franek | Well, my map is not from a sphere to a cell-complex, but from a cell-complex (morover, compact manifold) to the sphere. Does it not change the situation? | |
| Feb 10, 2011 at 15:56 | comment | added | Ryan Budney | Solving the word problem is exactly what you're asking to do. Checking if a map from a circle to a 2-complex is null homotopic is the word problem, just stated in a slightly different (but equivalent) form. So what you're looking to do is algorithmically impossible. You could of course write an algorithm that may fail infrequently, but it's guaranteed that there are examples it will get stuck on. The software such as GAP and Magnus do these kinds of things quite capably. | |
| Feb 10, 2011 at 0:42 | comment | added | Peter Franek | Thank you for your comment. However, I did not intend to compute the homotopy groups or so in general. The algorithm I hope for is much less: to check only for a given particular piece-wise linear map, whether it is homotopically trivial. And if it is trivial, whether in such case there exists a finite path of piecewise linear homotopies between some piece-wise linear maps, using only a finite set of such maps. But probably, even this is too much to hope :-( | |
| Feb 8, 2011 at 19:27 | history | edited | Ryan Budney | CC BY-SA 2.5 |
added 225 characters in body
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| Feb 8, 2011 at 19:07 | history | answered | Ryan Budney | CC BY-SA 2.5 |