We outline the main features of the definitions and applications of crossed complexes and cubical #-groupoids with connections.

Dedicated to Hvedri Inassaridze for his 70th birthday 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 computation of free crossed resolutions for amalgamated sums and HNN-extensions of groups, and so obtain computations of higher homotopical syzygies in these cases. 1

The classifying space of a crossed complex generalises the construction of Eilenberg-Mac Lane spaces. We show how the theory of fibrations of crossed complexes allows the analysis of homotopy classes of maps from a free crossed complex to such a classifying space. This gives results on the homotopy classification of maps from a CW-complex to the classifying space of a crossed module and also, more generally, of a crossed complex whose homotopy groups vanish in dimensions between 1 and n. The results are analogous to those for the obstruction to an abstract kernel in group extension theory.

Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with some historical background, and give a survey of the required basic facts on crossed complexes.

The aim is the theorems of the title and the corollary that the tensor product of two free crossed resolutions of groups or groupoids is also a free crossed resolution of the product group or groupoid. The route to this corollary is through the equivalence of the category of crossed complexes with that of cubical ω-groupoids with connections where the initial definition of the tensor product lies. It is also in the latter category that we are able to apply techniques of dense subcategories to identify the tensor product of covering morphisms as a covering morphism.

We prove that if M is a CW-complex and ∗ is a 0-cell of M, then the crossed module Π2(M,M 1, ∗) = (π1(M 1, ∗),π2(M,M 1, ∗),∂,⊲) does not depend on the cellular decomposition of M up to free products with Π2(D 2,S 1, ∗), where M 1 is the 1-skeleton of M. From this it follows that if G is a finite crossed module and M is finite, then the number of crossed module morphisms Π2(M,M 1, ∗) → G (which is finite) can be re-scaled to a homotopy invariant IG(M) (i. e. not dependent on the cellular decomposition of M), a construction similar to David Yetter’s in [Y], or Tim Porter’s in [P1, P2]. We describe an algorithm to calculate π2(M,M (1) , ∗) as a crossed module over π1(M (1) , ∗), in the case when M is the complement of a knotted surface in S 4 and M (1) is the 1-handlebody of a handle decomposition of M, which, in particular, gives a method to calculate the algebraic 2-type of M. In addition, we prove that the invariant IG yields a non-trivial

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