Talk:Desuspension
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Uhhh...
[edit]Shouldn't Σ^-1(Σ(X))=X? This is weird. Though, correct me if I'm wrong. GordyBeefPasta (talk) 09:30, 23 June 2024 (UTC)
Reworking the article
[edit]Most of the article is directly lifted from the first source, which is an invitation to an art workshop and clearly not trying to make precise mathematical statements. The citation of the second source is a chapter on triangulated categories, and doesn't seem relevant to the article (beyond mentioning the word desuspension in another context). This leads to most of the claims on the article being straight up wrong (or so imprecise they cannot be assigned a truth value), such as
- The "definition" section is so imprecise that it's not even clear what type of object Σ^{-1} X is supposed to be, though it seems to insinuate that it should be a space. (and when X is an n-dimensional manifold, it should be an (n-1)-dimensional manifold). This is not true and isn't claimed in the cited sources either, the desuspension should be a spectrum, see e.g. [Example 2.1.8 | https://people.math.binghamton.edu/malkiewich/spectra_book_2023Aug08.pdf].
- "Desuspension makes the category of spaces a triangulated category." This is not even possible, since triangulated categories are additive but the category of (pointed) topological spaces isn't, since its coproducts and products are distinct, even after descending to homotopy. In order for the statement to be true, one should talk about the stable homotopy category
- "If arbitrary coproducts were allowed, desuspension would result in all cohomology functors being representable." I don't understand at all what this is even trying to say? If arbitrary coproducts were allowed where?
Cleanup would involve rewriting the article to be about spectra and the left shift functor in the category of spectra / stable homotopy category. The contents of the first source article would work well for giving motivation/intuition for the concepts (but obviously not for any mathematical statements). J Marttinen (talk) 16:24, 24 May 2026 (UTC)