Algebraic topology of the Lagrange inversion

Abstract

The Lagrange inversion formula for power series is one of the classical formulas from analysis and combinatorics. A nice geometric interpretation of this formula in terms of the Stasheff polytopes was discovered by Loday. We show that it also admits a natural topological interpretation in terms of the Chern numbers of the complex projective space. The proof is based on our earlier work on the Chern-Dold character in complex cobordism theory and leads to a new derivation of the Lagrange inversion formula. We provide a similar interpretation of the multiplicative inversion formula in terms of Chern numbers of the smooth theta divisors. In this relation we introduce a new formal group defined by the Catalan numbers and explain the topological meaning of the corresponding Hirzebruch genus. Finally, we discuss a related general problem of when all Chern numbers of an algebraic variety are divisible by its Euler characteristic.