COMBINATORIAL HOMOTOPY. I

Abstract

This chapter discusses the theory of nuclei and n-groups and its relation to Reidemeister's Überlagerungen. It presents a new definition of n-groups, or n-types. This is stated in terms of (n–1)-homotopy types, which were introduced by R. H. Fox. The series of n-types (n = 1, 2, …) is a hierarchy of homotopy, and a fortiori of topological invariants. That is to say, if two complexes, K, L, are of the same n-type, then they are of the same m-type for any m<n, where n ≤ ∞ and the ∞-type means the homotopy type. If dim K, dim L ≤ n then K, L are of the same homotopy type if they are of the same (n+1)-type. Two complexes are of the same 2-type when, their fundamental groups are isomorphic. Moreover, any discrete group is isomorphic to the fundamental group of a suitably constructed complex. Therefore, the classification of complexes according to their 2-types is equivalent to the classification of groups by the relation of isomorphism. Thus, the n-type (n2) is a natural generalization of a geometrical equivalent of an abstract group.