homotopy group of a spectrum




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.


For pointed topological spaces

Given a pointed topological space XX, then 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

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.


Suspension isomorphism


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.


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) \,.

For Omega-spectra

For EE a (weak) Omega spectrum then the colimit is attained:

π n(E){π n(E 0) forn0 π 0(E n) forn0 \pi_n(E) \simeq \left\{ \array{ \pi_n(E_0) & for \; n \geq 0 \\ \pi_0(E_{-n}) & for \;n \leq 0 } \right.

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.




Last revised on September 22, 2018 at 05:57:31. See the history of this page for a list of all contributions to it.