In general a periodic map on a finite CW-complex leads to a periodic family of elements in $\pi^S_*$, although the procedure is not always as simple as in the above examples. Each of them has the following features. We have a CW-complex (defined below in A1.1) $X$ of type $n$ with bottom cell in dimension $k$ and top cell in some higher dimension, say $k+e$. Thus we have an inclusion map $i_0: S^k \to X$ and a pinch map $j_0: X \to S^{k+e}$.
From Ravenel, Douglas C., Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies. 128. Princeton, NJ: Princeton University Press. xiv, 209 p. (1992). ZBL0774.55001, pp. 17-18.
In this text, how are $i_0$ and $j_0$ defined? I googled but couldn't find a reference for this, and the book doesn't explain further.
Intuitively I'm guessing that $i_0$ is $S^k \to \amalg S^k \hookrightarrow X$ where the first map is obtained from the H-cogroup structure on $S^k$ (doesn't that assume that there are only finitely many $k$-cells?) and the second is $k$-skeleton inclusion; similarly, I guess that $j_0$ is $X \twoheadrightarrow \amalg S^{k+e} \to S^{k+e}$ where the first map is collapsing the $(k+e-1)$-skeleton and the second is the codiagonal. Is that correct?
The context of this assertion is a bit unclear, but as I read it, Ravenel is referring to the spaces in Example 2.4.1 (Toda-Smith complexes). In particular, these CW-complexes have the property that they have just one $k$-cell and just one $(k+e)$-cell. So there is nothing complicated going on with a cogroup structure or codiagonal: $i_0$ is just the inclusion of the entire $k$-skeleton of $X$ and $j_0$ is the quotient by the $(k+e-1)$-skeleton.