Algebraic topology of the Lagrange inversion

TL;DR

This paper offers a novel topological interpretation of the Lagrange inversion formula using Chern numbers and complex cobordism theory, connecting algebraic combinatorics with topology.

Contribution

It introduces a topological perspective on the Lagrange inversion formula and related identities via Chern numbers, formal groups, and cobordism theory.

Findings

Topological interpretation of Lagrange inversion via Chern numbers

Introduction of a formal group defined by Catalan numbers

Connection between Chern numbers divisibility and Euler characteristic

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.