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Given two oriented, smoothly-embedded copies of $S^3$ in $S^4$ (called Schoenflies spheres), one can take an oriented connect-sum of the pairs $(S^4, M_1) \# (S^4, M_2)$. This puts a monoidal structure on the isotopy classes of Schoenflies spheres, with the unit being the linearly embedded $S^3 \to S^4$.

I know only one proof that this monoid is a group in the literature. I'm wondering if there are any others?

i.e. I'm looking for a published proof that pre-dates 2019. I strongly suspect one exists, but if I've seen it, I've forgotten.

Bonus question (but does not need to be answered), are there any analogous results for the monoidal structure on smooth structures on $S^4$ using the oriented connect-sum operation? For this, I suspect the answer is no.

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    $\begingroup$ Wouldn't the question make more sense if you add references? You say that you know "only one proof that this monoid is a group in the literature" and ask for other proofs. Which proof do you know? Also what "analogous results for the monoidal structure on smooth structures on $S^4$" are you asking about? $\endgroup$
    – Igor Belegradek
    Commented Mar 24, 2023 at 14:01
  • $\begingroup$ After the second look I realized that you are probably asking at the end if the monoid of smooth structures on $S^4$ is a group. $\endgroup$
    – Igor Belegradek
    Commented Mar 24, 2023 at 14:52
  • $\begingroup$ @IgorBelegradek: I'm asking for a reference to a proof before 2019, i.e. I do not have references. You seem to want me to talk about what this question is specifically not about. I don't see the point. $\endgroup$
    – Ryan Budney
    Commented Mar 25, 2023 at 20:26
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    $\begingroup$ Which 2019+ proof exists? $\endgroup$
    – Chris Gerig
    Commented Mar 27, 2023 at 22:28
  • $\begingroup$ @ChrisGerig: if a proof is already in the literature, the purpose of this thread is to make people aware of it. If there isn't one (pre 2019) I'd maybe write a short expository argument that outlines how this follows with an elementary argument from published material. Given the lack of response in this thread I've contacted a few people directly, to verify if this result is known. $\endgroup$
    – Ryan Budney
    Commented Mar 29, 2023 at 0:18

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