nLab
homotopy group of a spectrum

Contents

Contents

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 XX, its stable homotopy groups are the colimit of ordinary homotopy groups of its reduced suspensions

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

For sequential spectra

Definition

Given a sequential spectrum EE, in the form of a sequence of component spaces E nE_n with structure maps ΣE nE n+1\Sigma E_n \to E_{n+1}, then for kk \in \mathbb{Z} the nnth homotopy group of EE is the colimit

π n(E) lim kπ n+k(E k) lim(π n+k(E k)Σπ n+k+1(ΣE k)π n+k+1(σ k E)π n+k+1(E k+1)) \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 XX is an Omega-spectrum, then its colimiting stable homotopy groups according to Def. reduce to the actual homotopy groups of the component spaces X nΩ Σ nXX_n \coloneqq \Omega^\infty \Sigma^n X, in that:

Xis Omega-spectrumπ k(X){π k+n(X n) | k+n0 π k(X 0) | k0 π 0X |k| | k<0 . 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

Σ:SeqSpecSeqSpec \Sigma \;\colon\; SeqSpec \longrightarrow SeqSpec

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

Proposition

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

S 1():π (X)π +1(ΣX). 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=Σ XE = \Sigma^\infty X the suspension spectrum of a pointed topological space, we have

π n S(X)π n(Σ X). \pi_n^S(X) \simeq \pi_n(\Sigma^\infty X) \,.

As a homology theory

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

Xπ st(X)π (Σ X) 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

Examples

References

Last revised on January 20, 2021 at 09:04:30. See the history of this page for a list of all contributions to it.