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The suspension and loop space functors, Σ and Ω, operate on the lattice of Bousfield classes of (sufficiently highly connected) topological spaces, and therefore generate a submonoid L of the complete set of operations on the Bousfield lattice. We determine the structure of L in terms of a single parameter of homotopy theory which is closely tied to the problem of desuspending weak cellular inequalities. Introduction. A topological space Y is said to be X-acyclic if the weak contractibility of the space of pointed maps from X to Z implies the weak contractibility of the maps from Y to Z, that is, if the implication map∗(X,Z) ∼ ∗ ⇒ map∗(Y,Z) ∼ ∗ holds for all spaces Z. This relation is denoted X < Y , and is referred to as a weak cellular inequality. Spaces X and Y are Bousfield equivalent if both X < Y and Y < X; the Bousfield equivalence class of X is denoted 〈X〉. Bousfield classes (of spectra) play an important role in stable homotopy theory, and are crucial to the development of unstable analogs of the nilpotence and periodicity results of Devinatz, Hopkins and Smith. Chachólski has shown [2] that the suspension and loop space functors Σ and Ω respect weak cellular inequalities; that is, if X < Y , then also ΣX < ΣY and ΩX < ΩY , provided X and Y are both path connected. Consequently, if 〈X〉 = 〈Y 〉, then 〈ΣX〉 = 〈ΣY 〉 and 〈ΩX〉 = 〈ΩY 〉. It is easily proved using the James reduced product construction that 〈ΣΩΣX〉 = 〈ΣX〉 for any space X. Additionally, it was observed in the 1950’s that a connected space X is an H-space if and only if it is a retract of ΩΣX. This implies that 〈ΩΣΩX〉 = 〈ΩX〉 for any connected space X. It is less obvious, however, that for every path connected space X, 〈ΣΩΣΩΣX〉 = 〈ΣX〉 2000 Mathematics Subject Classification: 55P60, 55P65, 55P99, 20M05.

The monoid of suspensions and loops modulo Bousfield equivalence