Stability of Loday constructions

Abstract

We study the question for which commutative ring spectra A the tensor of a simplicial set X with A, X ⊗ A, is a stable invariant in the sense that it depends only on the homotopy type of ΣX. We prove several structural properties about different notions of stability, corresponding to different levels of invariance required of X ⊗ A. We establish stability in important cases, such as complex and real periodic topological K-theory, KU and KO.