# nLab suspension spectrum

### Context

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Definition

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

$(\Sigma^\infty X)_n = \Sigma^n X$

## Properties

### Relation to looping and stabilization

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

$(\Sigma^\infty \dashv \Omega^\infty)\colon Top_* \stackrel{\overset{\Omega^\infty}{\leftarrow}}{\underset{\Sigma^\infty}{\to}} Spec$

(…)

## 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)