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Idea

Stable homotopy groups are homotopy groups as seen in stable homotopy theory.

A morphism inducing an isomorphism on all stable homotopy groups is called a stable weak homotopy equivalence.

Definition

For pointed topological spaces

Given a pointed topological space $X$, its stable homotopy groups are the colimit of ordinary homotopy groups of its reduced suspensions

For sequential spectra

\begin{definition} \label{StableHomotopyGroupsOfSequentialSpectrum}

Given a sequential spectrum $E$, in the form of a sequence of component spaces $E_n$ with structure maps $\Sigma E_n \to E_{n+1}$, then for $\mathrm{<span><del\; class="diffmod">\; k</del><ins\; class="diffmod">\; n</ins></span>}\in \mathbb{Z}$ k n \in \mathbb{Z} the $n$th homotopy group of $E$ is the colimit

over the homotopy groups of the component spaces.

\end{definition}

For sequential spectra in simplicial sets, the same formula applies for the geometric realization of the component simplicial sets.

(For details see this definition.)

\begin{remark} \label{StableHomotopyGroupsOfOmegaSpectrum}

If a sequential spectrum $X$ is an Omega-spectrum, then its colimiting stable homotopy groups according to Def. \ref{StableHomotopyGroupsOfSequentialSpectrum} reduce to the actual homotopy groups of the component spaces $X_n \coloneqq \Omega^\infty \Sigma^n X$, in that:

\end{remark}

(For details see this example.)

Properties

Suspension isomorphism

Let

be the operation of forming degreewise the smash product with the circle, the (un-derived, reduced) suspension of $X$.

(e.g. Schwede 12, part I, prop. 2.6)

For suspension spectra

For $E = \Sigma^\infty X$ the suspension spectrum of a pointed topological space, we have

As a homology theory

The assignment of stable homotopy groups to topological spaces $X$ (CW-complexes)

satisfies the axioms of a generalized homology theory. As such this is also called stable homotopy homology theory.

Examples

• stable homotopy groups of spheres

• Thom's theorem

Related concepts

• homotopy group

• free infinite loop space

• equivariant homotopy group

Examples

• The homotopy groups of a suspension spectrum $\Sigma^\infty X$ of a pointed topological space $X$ are the stable homotopy groups of $X$.

• In particular the homotopy groups of the sphere spectrum are the stable homotopy groups of spheres.

• Thom's theorem says that the homotopy groups of the Thom spectrum $M O$ form the (unoriented) cobordism ring.

References

• Frank Adams, part III, section 2 of Stable homotopy and generalised homology, 1974

• Stefan Schwede, Symmetric spectra (2012)