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?