Some topological genera and Jacobi forms.

We revisit and elucidate the Formula: see text-genus, Hirzebruch's Formula: see text-genus, and Witten's Formula: see text-genus, cobordism invariants of special classes of manifolds. After slight modification, involving Hecke's trick, we find that the Formula: see text-genus and Formula: see text-genus arise directly from Jacobi's theta function. For every Formula: see text we obtain exact formulas for the quasimodular expressions of Formula: see text and Formula: see text as "traces" of partition Eisenstein series Formula: see text which are easily converted to the original topological expressions. Surprisingly, Ramanujan defined twists of the Formula: see text in his "lost notebook" in his study of derivatives of theta functions, decades before Borel and Hirzebruch rediscovered them in the context of spin manifolds. In addition, we show that the nonholomorphic Formula: see text-completion of the characteristic series of the Witten genus is the Jacobi theta function avatar of the Formula: see text-genus.