nLab
suspension spectrum

Contents

Definition

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

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

(e.g. Elmendorf-Kriz-May, example 1.1)

As a symmetric spectrum: (Schwede 12, example I.2.6)

Properties

Completion to an Ω\Omega-spectrum

See at Omega spectrum – Completion of a suspension spectrum.

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

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 March 23, 2016 12:46:06 by Urs Schreiber (82.113.99.142)