Timeline for Explicit classifying spaces for crossed complexes
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 26, 2010 at 18:00 | vote | accept | Josh | ||
| Jan 25, 2010 at 17:27 | comment | added | Tim Porter | To be a bit more exact in my reply can I ask two things (i) How is the crossed complex being given to you? and (ii) what constitutes an explicit construction for you? In other words what sort of input and output have you in mind? As I said above, if you are happy with a simplicial set description and have `input' the crossed complex in a neat way, you can find the answer (sort of explicitly) in the Menagerie notes (see my n-Lab page) section 5.2.3. and if that does not give you an answer, please tell me as it means that the notes need improving! | |
| Jan 25, 2010 at 16:34 | comment | added | Tim Porter | I misunderstood your point, basically because your example was of a group. I did not quite understand what you meant by `the topology behind a certain group which fits into a truncated crossed complex'. Can you be a bit more explicit? You are right, what Ellis and Loday do is work out the crossed complex structure that corresponds to the classifying space of a group. That crossed complex will have the cell complex they construct as its classifying space (I think I am right). If you giev me a bit more information I will see what extra info I can think of. | |
| Jan 25, 2010 at 15:37 | comment | added | Josh | It looks like Ellis' papers that you're talking about (and Loday's too) are about classifying spaces of groups. Am I missing something? | |
| Jan 25, 2010 at 7:23 | comment | added | Tim Porter | Josh, The calculation you mention was done by Graham Ellis in his MSc thesis (way back, about 1982?). Graham has several early papers which contain parts of that stuff (I do not have MathSci access here at home so cannot do a search for you.) He also developed explicit constructions of classifying spaces using his software. (look at hamilton.nuigalway.ie/preprints/resolutions.pdf for instance.) The usual way is to see it both as a space AND as a crossed resolution in the context you seem to be studying. There are also constructions in Loday's paper on Homotopical Syzygies. | |
| Jan 24, 2010 at 23:58 | comment | added | Josh | Suppose $G=<t|t^m>$. Then $BG$ can be constructed inductively cell by cell by killing off higher homotopy and taking limit of the resulting spaces. It's (mostly) clear that the result is an infinite dimensional lens space. I would be happy seeing any example where someone says, here is a crossed complex, here is the construction, and here is the classifying space. | |
| Jan 24, 2010 at 16:50 | history | answered | Tim Porter | CC BY-SA 2.5 |