# Explicit classifying spaces for crossed complexes

I'm trying to understand the topology behind a certain group which fits into a truncated crossed complex, so I've been trying to understand Brown's construction of the classifying space of a crossed complex. I asked a similar question here and the answer cleared up some of my misunderstanding.

What I'm looking for now is an explicit construction of such a space. I've not found any papers that give explicit examples of such a construction, but hopefully there is one somewhere. I'm looking for something along the lines of the easy to follow of the construction of the classifying space of cyclic groups.

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## 1 Answer

It is not clear to me what you need / want. The classifying space of a cyclic group is constructed using a presentation and then killing off higher identities that may be around (there aren't any!). From that viewpoint the question you seem to ask is related to the combinatorial group theory of the group in question (or am I misderstanding the question.) There are examples that might help due to Loday in his paper on higher syzygies, but that may not quite fit the bill as he does not explicitly give the link with classifying spaces.

If you are happy with simplicial methods then you can build a simplicial T-complex from a crossed complex of groups by a modified Dold-Kan construction. The classifying space of that simplicial group (its Wbar) is something that has the same properties as Ronnie's classifying space. It is feasible if you know the crossed complex reasonably fully to construct this explicitly.

It depends what you need the construction for? How is your crossed complex arising precisely (and incidently what do you mean by the topology', will a simplicial model do)?

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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. – jd.r Jan 24 '10 at 23:58
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. – Tim Porter Jan 25 '10 at 7:23
It looks like Ellis' papers that you're talking about (and Loday's too) are about classifying spaces of groups. Am I missing something? – jd.r Jan 25 '10 at 15:37
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. – Tim Porter Jan 25 '10 at 16:34
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! – Tim Porter Jan 25 '10 at 17:27