Group-theoretic algebraic models for homotopy types

On simplicial resolutions of framed links

“…The Moore complex N G has a hypercrossed complex structure (see [5]) which admits the original simplicial group G to be rebuilt. By applying the simplicial structure to the link groups of the naive cabling of framed links, we have the following normal form theorem on these link groups.…”

3-types of simplicial groups and braided regular crossed modules

“…The fundamental idea behind these can be found in Carrasco and Ceggarra (cf. [12,13]). The construction depends on a variety of sources, mainly Conduché [16], Mutlu and Porter [24].…”

“…Carrasco and Cegarra [13] defined the notion of n-hypercrossed complex as an algebraic model of connected (n + 1)-types. The article [4] is one of a series in which the first author and Porter studied the higher dimensional Peiffer elements, called hypercrossed complex pairings F α,β , by using ideas of Conduché (cf.…”

“…[16]) and techniques developed by Carrasco and Cegarra (cf. [13]), and they applied their results in various homological settings and then gave a reformulation of Conduché's result in terms of hypercrossed complex pairings for commutative algebras. Mutlu and Porter [24] have also adapted their method to simplicial groups.…”

The fundamental 2-crossed complex of a reduced CW-complex

“…An equivalence Ψ between the categories of neat squared complexes and neat maps and 2-crossed complexes Definition 2.4. A 2-crossed complex [13,21] is given by a chain complex of groups…”