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The Nash-Tognoli theorem states that every closed and smooth manifold is diffeomorphic to a real algebraic variety. This appears to me as a very strong and surprising fact. However, I am not aware of any consequences, yet I have also been almost completely ignorant towards real algebraic geometry.

Question: What are nice consequences of the Nash-Tognoli theorem?

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I do not know about "nice" (which is a matter of taste) but one can prove (using the N-T theorem) that if $M$ is a smooth closed manifold, then there exists a polygonal linkage $L$ such that the moduli space $M(L)$ of planar realizations of $L$ has a component diffeomorphic to $M$. Bill Thurston named it "an airplane theorem." See

Kapovich, Michael; Millson, John J., Universality theorems for configuration spaces of planar linkages, Topology 41, No. 6, 1051-1107 (2002). ZBL1056.14077.

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  • $\begingroup$ Why "an airplane theorem"? Why "airplane"? I don't see the connection... Thanks for clarifying! $\endgroup$
    – Joseph O'Rourke
    Commented Jan 27, 2023 at 22:13
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    $\begingroup$ @JosephO'Rourke: Imagine you are flying in an airplane (as a passenger!) and you have a conversation with a person occupying a seat next to you. At some point, your neighbor asks you "What do you do for living?" You answer, "I am a mathematician, a topologist." The natural responce is something like "what's that?" or even "you mean a topographer?" What could you tell them? Thurston's answer would be along the lines "Let me give you an example. For instance, I can show how to build a bar-and-joint mechanism that can forge your signature." That's why it's an "airplane theorem." $\endgroup$
    – Misha
    Commented Jan 27, 2023 at 22:30
  • $\begingroup$ :-) ${}{}{}{}{}$ $\endgroup$
    – Joseph O'Rourke
    Commented Jan 28, 2023 at 1:11
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    $\begingroup$ Incidentally, Zach Abel proved in his Ph.D. thesis that the linkage can be truly planar in that bars do not cross one another. $\endgroup$
    – Joseph O'Rourke
    Commented Jan 28, 2023 at 1:14
  • $\begingroup$ @JosephO'Rourke: Thank you for reminding me the name! $\endgroup$
    – Misha
    Commented Jan 28, 2023 at 4:26
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While not relying on Nash-Tognoli, that theorem makes something like the Brieskorn spheres inevitable. Though it is still surprising how simple are the polynomials that cut out exotic $S^7$'s in $\mathbb{C}^5$.

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    $\begingroup$ There are also similarly simple polynomials in 5 variables which cut out nonsmoothable PL manifolds, see here. $\endgroup$
    – Moishe Kohan
    Commented Jan 28, 2023 at 4:39

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