12 - Computing homotopy types using crossed n-cubes of groups  pp. 187-210

Computing homotopy types using crossed <i>n</i>-cubes of groups

By Ronald Brown

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The aim of this paper is to explain how, through the work of a number of people, some algebraic structures related to groupoids have yielded algebraic descriptions of homotopy n-types. Further, these descriptions are explicit, and in some cases completely computable, in a way not possible with the traditional Postnikov systems, or with other models, such as simplicial groups.

These algebraic structures take into account the action of the fundamental group. It follows that the algebra has to be at least as complicated as that of groups, and the basic facts on the use of the fundamental group in 1-dimensional homotopy theory are recalled in Section 1. It is reasonable to suppose that it is these facts that a higher dimensional theory should imitate.

However, modern methods in homotopy theory have generally concentrated on methods as far away from those for the fundamental group as possible. Such a concentration has its limitations, since many problems in the applications of homotopy theory require a non-trivial fundamental group (low dimensional topology, homology of groups, algebraic K-theory, group actions, …). We believe that the methods outlined here continue a classical tradition of algebraic topology. Certainly, in this theory non- Abelian groups have a clear role, and the structures which appear arise directly from the geometry, as algebraic structures on sets of homotopy classes.

It is interesting that this higher dimensional theory emerges not directly from groups, but from groupoids.