#### Volume 17, issue 2 (2017)

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

### Jie Wu and Zihong Yuan

Algebraic & Geometric Topology 17 (2017) 895–915
##### 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 $F{\left({ℝ}^{n+1},j\right)}^{+}\phantom{\rule{0.3em}{0ex}}{\wedge }_{{\Sigma }_{j}}\phantom{\rule{0.3em}{0ex}}{X}^{\left(j\right)}$.

##### Keywords
Snaith splitting, iterated loop suspension, functorial homotopy decomposition, coalgebra-split sub-Hopf algebra
##### Mathematical Subject Classification 2010
Primary: 55P35
Secondary: 55P48, 55P65