# Stability of Loday constructions

## TL;DR

This paper investigates conditions under which the Loday construction for commutative ring spectra remains invariant under certain transformations, establishing stability results for key examples like complex and real K-theory.

## Contribution

It introduces structural properties of stability for Loday constructions and proves stability for important spectra such as KU and KO.

## Key findings

- Stability depends only on the homotopy type of the suspension of X.
- Established stability for complex and real periodic topological K-theory.
- Provided structural insights into different notions of stability.

## Abstract

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