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The Bernoulli numbers appear in the Harer-Zagier formula enumerating gluings of polygons, the Kervaire-Milnor formula for the order of homotopy groups for n-spheres, and (with the connection to the Riemann zeta function at positive integer values) Witten's formulas for volumes of curvature forms in moduli spaces for quantum guage theories in two dimensions. Hirzebruch used his apparently intimate knowledge of them and the Norlund genearlized (convolved) Bernoulli numbers in setting up his theory of genera, and Milnor and Stasheff devote an appendix to them in their book Characteristic Classes. Lie theory is tied up with the structure of manifolds, and the Bernoulli numbers enter Lie theory through the BCH theorem. So, there are many cameo appearances of the numbers in topology.

Who first introduced the Bernoulli numbers into topology and how?

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The earliest occurence Milnor mentions in his survey Differential Topology Forty-six Years Later is Whitehead's paper On the Homotopy Groups of Spheres and Rotation Groups (1942). If $J_n$ is the image of the stable Whitehead homomorphism from the $n$th stable homotopy group of rotation groups to the $n$th stable homotopy group of spheres, then $J_n$ is cyclic of order $B_k/4k$ for $n=4k-1$. Whitehead's paper is the source for the Milnor-Stasheff's appendix, and the starting point of Milnor's, Brumfiel's, Kervaire's, Browder's, etc. 1950-1960s results on homotopy groups of spheres, that was the boom age of the subject. Their work eventually led to computing the number of exotic spheres, homeomorphic but not diffeomorphic to the standard Euclidean sphere. The discovery of exotic spheres in 1956 is what earned Milnor his Fields medal in 1962. By the way, here is a video of Milnor delivering Hedrik lecture on exotic spheres back in 1965!

On Bernoulli numbers in general see Math Overflow thread Why do Bernoulli numbers arise everywhere? Sums of powers of integers can be expressed in their terms, as Faulhaber discovered for first 17 powers in 1631, before Bernoulli, and their generating function is $t/(e^t-1)=\sum_{n=0}^{\infty}\frac{B_n\;t^n}{n!}$. According to Henry Cohn "the reason why $t/(e^t-1)$ is a natural generating function to consider is that we sometimes want to invert $e^t-1$ (the factor of t is just to make it holomorphic), and the most important reason I know of to invert it is that we want to invert $\Delta=e^D-1$", where $D$ is the derivative and $\Delta$ is the finite difference. But this does not quite explain their appearance in topology.

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  • $\begingroup$ I suspect a connection might have been made earlier by Poincare in investigating divergent series, group theory, and manifolds. $\endgroup$ – Tom Copeland Oct 17 '15 at 0:23

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