The question is about understanding suspension and desuspension, see also a previous question.
Question: How do we define desuspension exactly? (Please see the comments below, people complain about the meanings of desuspension in Wikipedia is useless).
Are we able to have the desuspension acting on the topological space as the suspension does? Or do we only have the desuspension act on the spectra but not the space?
$$\Sigma^{-1}(\Sigma{X})\neq X,$$
What is the intuitive explanation?
What are some examples with $\Sigma^{-1}(\Sigma{X})= X,$ and counter examples $\Sigma^{-1}(\Sigma{X})\neq X$?
Naively, I thought that for $X=S^1$
$$\Sigma^n{X}=S^{n+1},$$
while
$$\Sigma^{-n}{S^{n+1}}=S^{1},$$ is this still correct?
Note add: The desuspension is arguably firstly introduced in the cited text mentioned in H. R. Margolis (1983). Spectra and the Steenrod Algebra. North-Holland. p. 454.