Stable functorial decompositions of $F(\mathbb{R}^{n+1},j)^{+}\wedge_{\Sigma_j}X^{(j)}$

Abstract

We first construct a functorial homotopy retract of ${\Omega }^{n+1}{\Sigma }^{n+1}X$ for each natural coalgebra-split sub-Hopf algebra of the tensor algebra. Then, by computing their homology, we find a collection of stable functorial homotopy retracts of $F{\left({ℝ}^{n+1},j\right)}^{+}{\wedge }_{{\Sigma }_j}{X}^{\left(j\right)}$.