Computability of homotopy groups of nilpotent complexes

Abstract

In 1956, E.H. Brown Jr. proved the following theorem: Let X be a connected, simply connected, locally finite, simplicial complex. Then n,X is effectively computable for n > 1. In this paper we generalize this theorem to the case where X is a nilpotent complex. (See the end of Section 1 for some examples.) We rely on the fact that the Postnikov system of a nilpotent complex has a principle refinement. As in the simply connected case, this system is recursively computable. However, the nilpotent case differs from the simply connected case in that the homotopy groups of X cannot be read directly from the Postnikov system. This requires the construction, for each n > 0, of an n-connected cover of X. Because this construction is less well controlled with respect to the computability of homology than the Postnikov system, the homotopy groups of X are recursively computable. That is, we give a procedure which generates a recursively enumerable abelian group presentation of n, X, whereas if n,X is a finite nilpotent group it is possible to obtain a finite presentation of Z, X, for n > 1. In Section 1 we are concerned with preliminaries, including the definitions of a recursive function, and a recursively enumerable (r.e.) presentation. In Section 2 the description is given of an inductive construction of the Postnikov system of a locally finite nilpotent connected complex. In this construction we adapt the method of Brown [l], with only the minor alterations necessary to obtain the refined Postnikov system appropriate for a nilpotent complex. Section 3 introduces the applications of some notions, borrowed from recursion theory, to simplicial topology. In particular, we define recursive simplicial sets and maps. We then show that such objects behave nicely with respect to such topological constructions as pullbacks and pathspaces. In addition we show that K(G; n) is a recursive complex when G is nicely presented. Our primary motivation for the adoption of these ideas is that the homology groups of a recursive complex are recursively