On the 3-Type of a Complex

Abstract

Publisher Summary 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 π 1 , the second homotopy group π 2 , and a certain three-dimensional co-homology class of π 1 in π 2 . It presents connected cell complexes K and denote by K n the n -dimensional skeleton of K . The classification of complexes according to their 2-type is equivalent, under the correspondence K → π 1 ( 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 K n , it may replace K by K 3 when it discusses the 3-type. Therefore, it is assumed that any given complex is at most 3-dimensional.