Understanding De/Suspension $\Sigma^{-1}(\Sigma{X})\neq X$

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?

$$\Sigma^{-1}(\Sigma{X})\neq X,$$

1. What is the intuitive explanation?

2. 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.

• What is the definition of desuspention. The link you have sited doesn't provide any good explanation. Aug 18 '18 at 21:09
• Desuspension is for spectra, not spaces. Aug 19 '18 at 1:13
• Dear all, I don't understand why the question is put on hold - I provide the information that one can find on the website. Aug 19 '18 at 23:27
• That wikipedia article is just awful, I would recommend that you ignore it. Aug 20 '18 at 0:12
• I'm actually the one voting to reopen, but "unclear what you're asking" seems not too unjustified. Let me infer from what you said that you do not know the precise definition of desuspension when posting this question. So the first question you should have asked is, "What does this wikipedia page mean? How do we define desuspension exactly?". None of these are shown in your question. Instead, it is worded like you've already been clear what desuspension is, and is merely asking for "intuition" and "examples". (cont) Aug 20 '18 at 1:12