ON THE 3-TYPE OF A COMPLEX

Abstract

The standard algebraic invariants of a topological space depend only on the homotopy type of the space. This chapter discusses with a part of the converse problem of the determination of the homotopy type by algebraic invariants, and shows in effect that the only one-and two-dimensional invariants that enter are the fundamental group π1, the second homotopy group π2, and a certain three-dimensional co-homology class of π1 in π2. It presents connected cell complexes K and denote by Kn the n-dimensional skeleton of K. The classification of complexes according to their 2-type is equivalent, under the correspondence K → π1(K), to the classification of groups by the relation of isomorphism. The chapter presents an algebraic equivalent of the 3-type. As the n-type of K depends only on Kn, it may replace K by K3 when it discusses the 3-type. Therefore, it is assumed that any given complex is at most 3-dimensional.