On the Lusternik-Schnirelmann Category of Abstract Groups

Abstract

Let H be an abstract group. The cohomology groups H'(1l, A) may then be considered for any integer q > 0 and any abelian group A with II as a group of operators. The least integer n such that H'(II, A) = 0 for all A and all q > n is called the dimension of I. If no such integer exists then dim II = 00. For given HI, one can construct a connected aspherical CW-complex K11 with H as fundamental group.' Any two such complexes have the same homotopy type. The least dimension of such a complex Ku is called the geometric dimension of H. It is always infinite if H contains elements of finite order. The (Lusternik-Schnirelmann) category of a topological space is the least integer n such that can be covered by open sets UO, * * *, Un such that Ui is contractible in X. If no such integer exists then cat = oo. If we replace the phrase UE is contractible in X by each closed path in Us is contractible in X we obtain the 1-dimensional category of (notation: cat, X).2 Both cat and cat1 are homotopy type invariants. The category of the aspherical complex Ku (which is independent of the choice the complex K11) is called the category of I. Since K11 is aspherical we have cat Ku = cat1 Ku1, there is therefore no need to define cat IH. The main result of this note is THEOREM 1. For any group HI