On Decompositions in Homotopy Theory

We first describe Krull-Schmidt theorems decomposing H spaces and simply-connected co-H spaces into atomic factors in the category of pointed nilpotent p-complete spaces of finite type. We use this to construct a 1-1 correspondence between homotopy types of atomic H spaces and homotopy types of atomic co-H spaces, and construct a split fibration which connects them and illuminates the decomposition. Various properties of these constructions are analyzed. The Krull-Schmidt property first arose in the theory of R-modules, and when valid, it states that each object decomposes in a unique way into a sum of indecomposable objects of the same type. Numerous examples of decomposing the loop space on a co-H space can be found in the literature ([Hi], [M], [CM], [AG], [G1], [G2]). Typically what happens is that the loop space of an atomic co-H space is a product of various factors, and the least connected factor is an H space of special interest, while the other factors, in some sense, represent noise. The first general Krull-Schmidt type theorem in homotopy theory was proved by Wilkerson [W] for the p-localization of simply-connected finite complexes which are either H spaces or co-H spaces. Various stable versions appear in [F], [Ma] and [H]. We will eliminate the finite complex assumption at the expense of retreating to the category C∧ p of pointed connected nilpotent p-complete spaces with Hi(X;Z/p) finitely generated for each i. Accordingly, we restrict ourselves to this category in the sequel. All colimit constructions will be completed without further notice. In particular, co-H spaces will be defined in terms of the coproduct (which is the completion of the one point union), suspensions will be completed and loop spaces will only be considered when the underlying space is simply connected. In section 1 we will exploit the strengthened notion of atomicity in this category developed by Adams and Kuhn [AK], and prove a Krull-Scmidt theorem (Theorem A below). Theorem A. Each H space in C∧ p is homotopy equivalent to the weak direct product of atomic H spaces unique up to order. Each simply-connected co-H space in C∧ p is homotopy equivalent to the coproduct of atomic complete co-H spaces unique up to order. In section 2 we give a general correspondence between retracts of n-fold suspensions and retracts of n-fold loop spaces (2.2). In particular, for n = 1 we get the following theorem. Received by the editors February 5, 2004. 2000 Mathematics Subject Classification. Primary 55P35; Secondary 55P30, 55P45. c ©2005 American Mathematical Society