Let $G$, $H$ be topological groups and let $t \colon G \to H$ be a homomorphism, such that $t \colon G \to H$ is a topological crossed module. For a topological space $X$ we can define the nonabelian cohomology set $\check{H}^1(X, G \to H)$. There is a map of crossed modules from $1 \to H$ to $G \to H$ and this induces a map $$ \check{H}^1(X, H) \to \check{H}^1(X,G \to H) $$ What are the conditions for this map to be injective? If I express this problem in terms of classifying spaces, then I think I am asking for the fiber of the map $$ BH \to B(G \to H)\ . $$ In particular, my vague hope was that if $G$ is contractible, then the above map is injective. Is this true?
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asked Jun 21, 2011 at 17:49
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$\begingroup$ (i) Would perhaps the homotopy fibre of the map be more informative? (ii) There are results in Larry Breen's work on Bitorsors that give some interpretation of what the elements of the non-Abelian cohomology with coefficients in a crossed module 'look like'. That source may give you the answer. (I have a summary of some of it in one of the versions of the Menagerie, but it is better to look at the original.) $\endgroup$– Tim PorterJun 21, 2011 at 18:17
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$\begingroup$ @Tim: I meant the homotopy fiber, sorry. $\endgroup$– Ulrich PennigJun 21, 2011 at 20:06
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2 Answers
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For results on the classifying space of (discrete) crossed modules, or more generally, crossed complexes, and in relation to homotopy classification and fibrations, see R. Brown, Exact sequences of fibrations of crossed complexes, homotopy classification of maps, and nonabelian extensions of groups, J. Homotopy and Related Structures 3 (2008) 331-343.
However the topological case has not been worked up in this format, as far as I know.
answered Jun 22, 2011 at 10:01
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1$\begingroup$ The recent paper of Murray, Roberts and Stevenson (arxiv.org/PS_cache/arxiv/pdf/1102/1102.4388v1.pdf) may be relevant to this. In any case it deserves a mention. :-) $\endgroup$ Jun 22, 2011 at 11:56
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The fiber of that map is well known to be $BG$, so what you expect is true.
answered Jun 21, 2011 at 18:16
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$\begingroup$ Any reference for that other than that it is well known :-). $\endgroup$ Jun 21, 2011 at 19:07
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$\begingroup$ The reference depends on the definition of classifying space you have in mind. Probably the paper where you've got your definition from would contain that result. Otherwise just tell me which classifying space you mean and I'll try to find a reference for that model. $\endgroup$ Jun 21, 2011 at 23:28