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In K-theory we have the Todd class and the $\hat A$ class.

The Todd class is named after the Cambridge geometer John Arthur Todd.

Where does the name $\hat A$ come from? Does the A stand for Atiyah?

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    $\begingroup$ I have no idea, but if it $A$ stands for Atiyah, then $\hat A$ should come from sir Michael Atiyah wearing a hat. $\endgroup$ May 10, 2019 at 20:41

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It seems that the $\hat{A}$ genus was first introduced in Section 23 of the paper "Characteristic Classes and Homogeneous Spaces, II", by Borel and Hirzebruch (1959). It is presented as a small modification of some $A$-genus previously introduced by Hirzebruch in his book "Neue Topologische Methoden in der Algebraischen Geometrie" (1956) (the $A$-genus is the genus associated to the power series $2z^{1/2}/sh(2z^{1/2})$, whereas the $\hat{A}$-genus is associated to the power series $(z^{1/2}/2)/sh(z^{1/2}/2)$). It is a resonable explanation of the hat on $\hat{A}$: first some $A$-genus was introduced, then some small modification $\hat{A}$, and it was only gradually realized that the $\hat{A}$-genus was more important that the $A$-genus (to the point that today I don't know if anyone still uses the terminology $A$-genus). It seems unclear if there is a good explanation for the $A$ in $A$-genus except that $A$ is a quite common letter. When Hirzebruch used the letter $A$ in his book published in 1956, I don't think that Atiyah was already thinking about these questions and so it seems very unlikely that $A$ stand for Atiyah.

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  • $\begingroup$ Very nice. I'd wondered if there was an $A$-genus as well as an $\hat A$-genus. $\endgroup$ May 11, 2019 at 4:12
  • $\begingroup$ Do you have any guess what the L in $L$-genus stands for? $\endgroup$ May 11, 2019 at 4:13
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    $\begingroup$ For the record, $A$-genus is already in (1954). I wonder if it was to suggest the arithmetic genus Hirzebruch had once denoted $\Pi(V)$; for he writes in the first lines of (1992): $\endgroup$ May 12, 2019 at 2:30
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    $\begingroup$ “The word “genus” is meant in the sense of my book (1956): A genus is a homomorphism of the Thom cobordism ring (...) into the complex numbers. Fundamental examples are the signature and the $\smash{\hat A}$-genus. The $\smash{\hat A}$-genus equals the arithmetic genus of an algebraic manifold, provided the first Chern class of the manifold vanishes. According to Atiyah and Singer it is the index of the Dirac operator on a compact Riemannian manifold with spin structure.” $\endgroup$ May 12, 2019 at 2:31
  • $\begingroup$ Nice finds Francois Ziegler -- thanks. $\endgroup$ May 14, 2019 at 21:26

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