Algebraic & Geometric Topology

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

Jie Wu and Zihong Yuan

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Abstract

We first construct a functorial homotopy retract of Ωn+1Σn+1X 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(n+1,j)+ ΣjX(j).

Article information

Source
Algebr. Geom. Topol., Volume 17, Number 2 (2017), 895-915.

Dates
Received: 9 September 2015
Revised: 18 April 2016
Accepted: 30 September 2016
First available in Project Euclid: 19 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1508431448

Digital Object Identifier
doi:10.2140/agt.2017.17.895

Mathematical Reviews number (MathSciNet)
MR3623676

Zentralblatt MATH identifier
1361.55011

Subjects
Primary: 55P35: Loop spaces
Secondary: 55P48: Loop space machines, operads [See also 18D50] 55P65: Homotopy functors

Keywords
Snaith splitting iterated loop suspension functorial homotopy decomposition coalgebra-split sub-Hopf algebra

Citation

Wu, Jie; Yuan, Zihong. Stable functorial decompositions of $F(\mathbb{R}^{n+1},j)^{+}\wedge_{\Sigma_j}X^{(j)}$. Algebr. Geom. Topol. 17 (2017), no. 2, 895--915. doi:10.2140/agt.2017.17.895. https://projecteuclid.org/euclid.agt/1508431448


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References

  • A Bahri, M Bendersky, F,R Cohen, S Gitler, The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces, Adv. Math. 225 (2010) 1634–1668
  • C-F Bödigheimer, Stable splittings of mapping spaces, from “Algebraic topology” (H,R Miller, D,C Ravenel, editors), Lecture Notes in Math. 1286, Springer, Berlin (1987) 174–187
  • F,R Cohen, The homology of $\mathcal{C}_{n+1}$–spaces, $n \geqslant 0$, from “The homology of iterated loop spaces” (F,R Cohen, T,J Lada, J,P May, editors), Lecture Notes in Mathematics 533, Springer, Berlin (1976) 207–351
  • F,R Cohen, J,P May, L,R Taylor, Splitting of certain spaces $CX$, Math. Proc. Cambridge Philos. Soc. 84 (1978) 465–496
  • F,R Cohen, J,P May, L,R Taylor, Splitting of some more spaces, Math. Proc. Cambridge Philos. Soc. 86 (1979) 227–236
  • N Dobrinskaya, Loops on polyhedral products and diagonal arrangements, preprint (2009)
  • J,Y Li, F,C Lei, J Wu, Module structure on Lie powers and natural coalgebra-split sub-Hopf algebras of tensor algebras, Proc. Edinb. Math. Soc. 54 (2011) 467–504
  • J,P May, L,R Taylor, Generalized splitting theorems, Math. Proc. Cambridge Philos. Soc. 93 (1983) 73–86
  • J McCleary, A user's guide to spectral sequences, 2nd edition, Cambridge Studies in Advanced Mathematics 58, Cambridge University Press (2001)
  • P Selick, J Wu, On natural coalgebra decompositions of tensor algebras and loop suspensions, Mem. Amer. Math. Soc. 701, Amer. Math. Soc., Providence, RI (2000)
  • P Selick, J Wu, On functorial decompositions of self-smash products, Manuscripta Math. 111 (2003) 435–457
  • V,P Snaith, A stable decomposition of $\Omega \sp{n}S\sp{n}X$, J. London Math. Soc. 7 (1974) 577–583
  • Z,H Yuan, Functorial homotopy decompositions of iterated loop suspensions, PhD thesis, National University of Singapore (2014) \setbox0\makeatletter\@url http://scholarbank.nus.edu.sg/bitstream/handle/10635/118255/Yuan_Zihong_PhD_Thesis.pdf {\unhbox0