suspension spectrum



For XX a pointed topological space, its suspension spectrum Σ X\Sigma^\infty X is the 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


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



  • 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 November 24, 2015 17:31:07 by Urs Schreiber (