nLab
suspension spectrum

Contents

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

Strong monoidalness

The suspension spectrum functor is strong monoidal.

On the one hand, this is the case for its incarnation as a 1-functor with values in structured spectra (this Prop.) Via the corresponding symetric monoidal model structure on structured spectra this exhibits strong monoidalness also as an (infinity,1)-functor.

More abstractly this follows from general properties of stabilization when regarding stable homotopy theory as the result of inverting smash product with the circle, via Robalo 12, last clause of Prop. 4.1 with last clause of Prop. 4.10 (1). For emphasis see also Hoyois 15, section 6.1, specifically Hoyois 15, Def. 6.1.

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)

Last revised on January 3, 2019 at 12:22:55. See the history of this page for a list of all contributions to it.