We introduce a pairing structure within the Moore complex NG of a simplicial group G and use it to investigate generators for N G n ∩ D n where D n is the subgroup generated by degenerate elements. This is applied to the study of algebraic models for homotopy types.
Generalising a result of Brown and Loday, we give for n = 3 and 4, a decomposition of the group, d n NG n , of boundaries of a simplicial group G as a product of commutator subgroups. Partial results are given for higher dimensions. Applications to 2-crossed modules and quadratic modules are discussed.
The following partition problem was first introduced by R.C. Entringer and has subsequently been studied by the first author and more recently by Bollobás and Scott, who consider the hypergraph version as well, using a probabilistic technique. The partition problem is that of coloring the vertex set of a graph with s colors so that the number of induced… (More)
In this paper we give a construction of free 2-crossed modules. By the use of a `step-by-step' method based on the work of Andr e, we will give a description of crossed algebraic models for the steps in the construction of a free simplicial resolution of an algebra. This involves the introduction of the notion of a free 2-crossed module of algebras.
Using free simplicial groups, it is shown how to construct a free or totally free 2-crossed module on suitable construction data. 2-crossed complexes are introduced and similar freeness results for these are discussed.
We introduce a notion of join for (augmented) simplicial sets gen-eralising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial category ∆.
Following Ellis, , we investigate the notion of totally free crossed square and related squared complexes. It is shown how to interpret the information in a free simplicial group given with a choice of CW-basis, interms of the data for a totally free crossed square. Results of Ellis then apply to give a description in terms of tensor products of crossed… (More)
The category of crossed complexes gives an algebraic model of CW-complexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating non-abelian extensions. We show how the strong properties of this category allow for the… (More)