Stable homotopy groups of a sequential prespectrum

Fix $k\in \mathbb{Z}$ and choose $n_0\ge 0$ such that $n+k\ge 1$ whenever $n\ge n_0$. For a sequential prespectrum $E$, the bonding homomorphism is

$\begin{array}{rl}{\displaystyle b_n:{\pi }_{n+k}(E_n)}& {\displaystyle ⟶{\pi }_{n+k+1}(E_{n+1}),}\\ {\displaystyle [f]}& {\displaystyle ⟼[{\sigma }_n\circ (1_{S^1}\wedge f)\circ \chi ],}\end{array}$

where $\chi :S^{n+k+1}\cong S^1\wedge S^{n+k}$ is the canonical sphere-coordinate homeomorphism. Equivalently, $b_n$ is suspension followed by $({\sigma }_n{)}_{\ast }$.

The $k$th stable homotopy group of $E$ is

${\pi }_k(E)={\mathrm{colim}}_{n\ge n_0}({\pi }_{n+k}(E_n),b_n)\mathrm{.}$

Concretely this is the disjoint union of the stage groups modulo $(n,x)\sim (n+1,b_{n}x)$. Two representatives are added after advancing both to a common stage. This is well defined because the $b_n$ are homomorphisms. After increasing $n_0$ if necessary, all degrees $n+k$ are at least two, so all stage groups and the resulting colimit are abelian. The next lemma proves that the displayed group does not depend on the chosen finite initial cutoff.