Algebraic topology Stable homotopy theory and generalized cohomology Object

Sphere spectrum

Builds on Spectrum Stable homotopy groups of spheres Stable homotopy category

Description

The sphere spectrum $S$ is the spectrum with $S_n = S^n$ , the $n$ -sphere, for $n \ge 0$ , and with structure maps the canonical homeomorphisms $\Sigma S^n \cong S^{n+1}$ . It is the suspension spectrum $\Sigma^\infty S^0$ of the two-point space. Its homotopy groups are the stable homotopy groups of spheres: $\pi_k(S) = \operatorname{colim}_n \pi_{k+n}(S^n) = \pi_k^s.$

Remarks

One has $\pi_k(S) = 0$ for $k < 0$ , $\pi_0(S) = \Z$ , $\pi_1(S) = \Z/2$ , $\pi_2(S) = \Z/2$ and $\pi_3(S) = \Z/24$ . For $k > 0$ every group $\pi_k(S)$ is finite, by Serre's finiteness theorem. In the stable homotopy category, $S$ is the unit for the smash product, and maps $S \to E$ of degree $k$ compute $\pi_k(E)$ for every spectrum $E$ . The multiplication $S^n \wedge S^m \cong S^{n+m}$ makes $S$ a commutative ring spectrum, and $\pi_*(S)$ is a graded commutative ring.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Spectrum concept
Stable homotopy category concept
Stable homotopy groups of spheres concept
Sphere spectrum object

Where this leads

3 pages build on this one.

Tag	`14CGGHA85T`
URI	`m2://14CGGHA85T`
Kind	object
Writing	W0 stub
MSC	55P42, 55Q45
Instance of	Spectrum
Uses	Spectrum, Stable homotopy groups of spheres, Stable homotopy category
Same as	Wikidata Q7576683 Wikipedia Sphere spectrum
Changed	2 November 2026

Sphere spectrum

Description

Remarks

Reading path

Where this leads

Further reading

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