Crossed complexes have longstanding uses, explicit and implicit, in homotopy theory and the cohomology of groups. It is here shown that the category of crossed complexes over groupoids has a symmetric monoidal closed structure in which the internal Hom functor is built from morphisms of crossed complexes, nonabelian chain homotopies between them and similar higher homotopies. The tensor product involves non-abelian constructions related to the commutator calculus and the homotopy addition lemma. This monoidal closed structure is derived from that on the equivalent category of ω-groupoids where the underlying cubical structure gives geometrically natural definitions of tensor products and homotopies.

Abstract. In this paper, we initiate the study of C ∗-algebras A endowed with a twisted action of a locally compact Abelian Lie group G, and we construct a twisted crossed product A⋊G, which is in general a nonassociative, noncommutative, algebra. The duality properties of this twisted crossed product algebra are studied in detail, and are applied to T-duality in Type II string theory to obtain the T-dual of a general principal torus bundle with general H-flux, which we will argue to be a bundle of noncommutative, nonassociative tori. We also show that this construction of the T-dual includes all of the special cases that were previously analysed. 1.

Dedicated to the memory of Frank Adams

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

Abstract. This is a survey on the use of some internal higher categorical structures in algebraic topology and homotopy theory. After providing a general view of the area and its applications, we concentrate on the algebraic modelling of connected (n + 1)-types through cat n-groups. 1.

A result describing the fundamental group of a union of two connected subcomplexes of a simplicial complex goes back at least to Seifert [14], in the case when the intersection is connected. This is an early result on the fundamental group of the union of two spaces. For Seifert, this result was a technical tool for describing the fundamental groups of 3-manifolds he constructed in various ways, in particular in terms of a Heegard decomposition. In modern language, Seifert’s result yields the fundamental group under discussion in terms of a pushout diagram of groups. Given an algebraic curve in the complex projective plane, using purely algebro-geometric methods, Zariski wrote down a presentation of the fundamental group of the complement and suggested to van Kampen to confirm the correctness of the presentation by purely topological methods, which he did in [17]. Thereafter van Kampen established a general theorem on the fundamental group of certain pathwise connected topological spaces [16]. This result underlies the contents of his previous paper; it is essentially the same as that established by Seifert for simplicial complexes. The problem with the connectivity assumption of the intersection prevented the use of the theorem for deducing the result that the fundamental group of the circle is free cyclic. In [1], R. Brown could then overcome this obstacle by generalizing the statement of the theorem from the fundamental group on one base point to the

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