Combinatorial homotopy. I

Introduction. This is the first of a series of papers, whose aim is to clarify the theory of "nuclei" and "w-groups" and its relation to Reidemeister's 1 berlagerungen. Here we give a new definition of "^-groups, " or n-types as we now propose to call them. This is stated in terms of (-l) -homotopy types, which were introduced by R. H. Fox. The series of w-types (w = l, 2, ) is a hierarchy of homotopy, and a fortiori of topological invariants. That is to say, if two complexes, 3 K, L, are of the same w-type, then they are of the same ra-type for any m2) is a natural generalization of a geometrical equivalent of an abstract group. 4 Following up this idea we look for a purely algebraic equivalent of an w-type when n > 2. An important requirement for such an algebraic system is "realizability, " in two senses. In the first instance this means that there is a complex which is in the appropriate relation to a given one of these algebraic systems, just as there is a complex whose fundamental group is isomorphic to a given group. The second kind, whose importance is underlined by theorems in 5; 6 and in this paper, is the "realizability" of homomorphisms, chain mappings, etc. , by maps of the corresponding complexes. Thus realizability means that the algebraic representation is not subject to conditions which can only be expressed geometrically. An address delivered before the Princeton Meeting of the Society on November 2, 1946, by invitation of the Committee to Select Hour Speakers for Eastern Sectional Meetings; received by the editors July 19, 1948. 1 See l, 3 and 8, p. 177, Numbers in brackets refer to the references cited at the end of the paper. 2 See 9, p. 343 and 10, p. 49. 3 I. e. , CW-complexes, as defined in 5 below. 4 I. e. , the class of groups which are isomorphic to a given group.