Centric Maps and Realization of Diagrams in the Homotopy Category

Let D be a small category. Suppose that X̄ is a D-diagram in the homotopy category (in other words, a functor from D to the homotopy category of simplicial sets). The question of whether or not X̄ can be realized by a D-diagram of simplicial sets has been treated by [5]. The purpose of this note is to study a special situation in which the treatment can be simplified quite a bit. We look at two examples to which this simplified treatment is applicable; both examples involve homotopy decomposition diagrams for compact Lie groups. Our results show that in at least one of these examples ([13]) the decomposition diagram is completely determined by its underlying diagram in the homotopy category. It is possible that this “rigidity” result will eventually contribute to a general homotopy theoretic characterization theorem for classifying spaces of compact Lie groups (cf. [8]). Before going any further we have to introduce some notation. The symbol S will denote the category of simplicial sets and hoS the associated homotopy category obtained by localizing with respect to (i.e. formally inverting) weak equivalences. If C is a category and D is a small category, then C will stand for the category of D-diagrams in C; the objects of C are functors D→ C and the maps of C are natural transformations. There is a projection functor π : S→ hoS; we will use the same symbol for induced functors S → (hoS). Both S and S admit closed simplicial model category structures [19, II] [20, p. 233] [4, §2] , and we will sometimes require without loss of generality that chosen objects in these categories be fibrant. Given a small category D and a diagram X̄ in (hoS), a realization of X̄ [5, 3.1] is a pair (Y, f) such that Y is an object of S and f is an isomorphism f : πY ∼= X̄ in (hoS). A weak equivalence between two