# nLab homotopy group of a spectrum

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# 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 $X$, then 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

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.

## 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 $X$

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

### For suspension spectra

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

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

### For Omega-spectra

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

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

### As a homology theory

The assignment of stable homotopy groups to topological spaces $X$ (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.