nLab homotopy group of a spectrum

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

$$\pi_n^S(X) \coloneqq \underset{\longrightarrow}{\lim}_k \pi_{n+k}(\Sigma^k X) \,.$$

For sequential spectra

Definition

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 $k \in \mathbb{Z}$ the $n$th homotopy group of $E$ is the colimit

$$\begin{aligned} \pi_n(E) & \coloneqq \underset{\longrightarrow}{\lim}_k \pi_{n+k}(E_k) \\ & \coloneqq \underset{\longrightarrow}{\lim} \left( \cdots \to \pi_{n+k}(E_k) \stackrel{\Sigma}{\longrightarrow} \pi_{n+k+1}(\Sigma E_k) \stackrel{\pi_{n+k+1}(\sigma_k^E)}{\longrightarrow} \pi_{n+k+1}(E_{k+1}) \to \cdots \right) \end{aligned}$$

over the homotopy groups of the component spaces.

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

(For details see this definition.)

Remark

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

$$X \; \text{is Omega-spectrum} \;\;\;\;\; \Rightarrow \;\;\;\;\; \pi_k(X) \simeq \left\{ \array{ \pi_{k+n}\big( X_n \big) &\vert& k + n \geq 0 \\ \pi_k\big(X_0\big) &\vert& k \geq 0 \\ \pi_0 X_{\vert k \vert} &\vert& k \lt 0 \\ } \right. \,.$$

(For details see this example.)

Properties

Suspension isomorphism

Let

$$\Sigma \;\colon\; SeqSpec \longrightarrow SeqSpec$$

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

Proposition

For $X$ a sequential spectrum, smashing with $S^1$ constitutes natural isomorphisms of stable homotopy groups of $X$ with the stable homotopy groups in one degree higher of the suspension spectrum of $X$

$$S^1 \wedge (-) \;\colon\; \pi_\bullet(X) \stackrel{\simeq}{\longrightarrow} \pi_{\bullet+1}(\Sigma X) \,.$$

(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

$$\pi_n^S(X) \simeq \pi_n(\Sigma^\infty X) \,.$$

As a homology theory

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

$$X \mapsto \pi_\bullet^{st}(X) \coloneqq \pi_\bullet(\Sigma^\infty X)$$

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)