Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $G_{\bullet}$ be a simplical group. I denote by $D_n \subset G_n$ the subgroup generated by all the degeneracy maps $s_i:G_{n-1} \to G_n$. I also denote $$M_n = \bigcap_{i>0} \mathrm{Ker} d_i \subset G_n$$ and $$Z_n = \bigcap_{i\geq 0} \mathrm{Ker} d_i \subset G_n$$ When $G_{\bullet}$ is a simplicial abelian group it is well known and there many references to the fact that

$$M_n \cap D_n = \{e\}$$

I know how to prove it for the non-abelian case, but I believe there should be a reference somewhere. In fact I'm actually interested in the weaker claim that :

$$Z_n \cap D_n = \{e\}$$

share|improve this question

1 Answer 1

up vote 5 down vote accepted

Tomer: The claim you make about $M_n\cap D_n$ being trivial in the non abelian case is not true. In fact that condition is equivalent to the Moore complex of $G$ being a crossed complex in the sense of Brown and Higgins. This seems to be first proved in the thesis of Ashley (N. Ashley, Simplicial T-Complexes: a non abelian version of a theorem of Dold-Kan, Dissertations Math., 165, (1989), 11 – 58). It is also a lemma in one of the two papers by Brown and Loday that if $G_2=D_2$, so there are no new non-degenerate elements in $G_2$, just ones there because of $G_1$, then $\partial(M_2\cap D_2)=[Ker d_0,Ker d_1]$. This condition implies that $M_1\to M_0$ is a crossed module. The kernel of that crossed module will not be trivial in general but is $Z_2\cap D_2$.

I think you have missposed the question (in its present form) and that there is some extra condition that you have omitted. (It is also possible that I have misunderstood exactly what you are asking!) To me it seems that the homotopy types that you can model with simplicial groups having your first condition would have vanishing Whitehead products and thus be products of Eilenberg-MacLane spaces.

As to a reference for this stuff, apart from the original sources, discussions of this area can be found in two of the draft texts to be found via my n-Lab pages (Menagerie and Profinite algebraic homotopy). They are also in papers of Pilar Carrasco and Antonio Cegarra (1991 JPAA).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.