Homotopy resolutions of semi-simplicial complexes

Abstract

One of the major problems of homotopy theory concerns the classification of topological spaces into homotopy types. This problem has as an obvious generalization that of the classification of spaces with operators into equivariant homotopy types. We make here a contribution toward the solution of the latter problem. Our treatment stems from those of Eilenberg and MacLane, Postnikov, Olum and Zilber, in that it uses the singular complex of a topological space in order to restate topological problems as combinatorial ones. But as Eilenberg and Zilber have pointed out, singular complexes, considered as algebraic structures, may be subsumed in the more general category of semi-simplicial complexes [3 ]. Our procedure will be to drop completely all topological structure and to consider the classification as a purely combinatorial problem in the domain of these semi-simplicial complexes. Thus the theory developed here is completely independent of topology. It is however almost completely parallel to the usual homotopy theory, and many of the same theorems will be found in it, though the proofs are often quite different. The treatment is divided into four parts. Chapter I develops, after a brief resume of the fundamental facts about semi-simplicial complexes, the notion of semi-simplicial complexes with operators. The operators envisaged are the cognates in the combinatorial domain of topological groups. Thus the complexes with operators correspond to spaces with topological groups of operators. In the most important cases the group operates without fixed points; these complexes are to be thought of as analogous to principal fibre bundles. Chapter II introduces the notion of homotopy groups of a semi-simplicial complex. These are to be used as coefficient domains for obstruction cochains, obstruction theory being the principal tool in the rest of the paper. The method here is quite unlike that used in the topological case; homotopy groups are characterized, simultaneously with obstructions, by a system of axioms. It is not however true that they can be defined for any semi-simplicial complex. Those complexes which have homotopy groups form a subcategory, and it is in this subcategory (which includes singular complexes) that the classification theory operates.