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So I came across the so-called Atiyah–Singer Index Theorem (ASIT) and claims of it being an extremely powerful and versatile tool.

Question. What is a truly simple application of the ASIT to obtain a nontrivial result, which (i.e the application) requires the least amount of specialized mathematics?

Thus, ideally the problem should only include basic tools from analysis, algebra (finite linear algebra, group theory), topology. Just elementary stuff.

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    $\begingroup$ Probably the Gauss-Bonnet theorem. Or is this not nontrivial enough? $\endgroup$
    – Wojowu
    Commented Jan 8, 2022 at 22:28
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    $\begingroup$ two examples are described here: math.stackexchange.com/a/815516/87355 $\endgroup$
    – Carlo Beenakker
    Commented Jan 8, 2022 at 22:28
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    $\begingroup$ Jon Hillman has an argument using Atiyah-Singer that excludes certain 3-manifolds from smoothly embedding in $S^4$. As far as I am aware, it is the only available argument, but this result might be derivable from some Heegaard-Floer machinery, not too far into the future. $\endgroup$
    – Ryan Budney
    Commented Jan 8, 2022 at 23:29
  • $\begingroup$ @RyanBudney: which argument are you referring to? Is it published? $\endgroup$
    – Marco Golla
    Commented Jan 9, 2022 at 12:41
  • $\begingroup$ He has a few papers. The first is: J.S.Crisp, J.A.Hillman: Embedding Seifert fibred 3-manifolds and Sol-manifolds in 4-space, Proceedings of the London Mathematical Society, 76 (1998), 685–710. If I recall the others I'll add another comment. $\endgroup$
    – Ryan Budney
    Commented Jan 9, 2022 at 20:58

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