nLab
suspension spectrum

Contents

Definition

For XX a pointed topological space, its suspension spectrum is the spectrum Σ X\Sigma^\infty X whose degree-nn space is the nn-fold reduced suspension of XX:

(Σ X) n=Σ nX (\Sigma^\infty X)_n = \Sigma^n X

Properties

Relation to looping and stabilization

As an infinity-functor Σ :Top *Spec\Sigma^\infty\colon Top_* \to Spec the suspension spectrum functor exhibits the stabilization of Top.

(Σ Ω ):Top *Σ Ω Spec (\Sigma^\infty \dashv \Omega^\infty)\colon Top_* \stackrel{\overset{\Omega^\infty}{\leftarrow}}{\underset{\Sigma^\infty}{\to}} Spec

Recognition and diagonals

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References

  • Nicholas J. Kuhn, Suspension spectra and homology equivalences, Trans. Amer. Math. Soc. 283, 303–313 (1984) (JSTOR)

  • John Klein, Moduli of suspension spectra (arXiv:math/0210258, MO)

Suspension spectra of infinite loop spaces are discussed (in a context of Goodwillie calculus and chromatic homotopy theory) in

  • Nicholas J. Kuhn, section 6.2 of Goodwillie towers and chromatic homotopy: An overview (pdf)

Revised on February 6, 2014 15:57:06 by Urs Schreiber (89.204.138.218)