Algebraic topology of the Lagrange inversion
Victor M. Buchstaber, Alexander P. Veselov
TL;DR
The paper develops a topological interpretation of the Lagrange inversion formula by leveraging the Chern-Dold character in complex cobordism to reinterpret its coefficients as monomial Chern numbers of CP^n and of theta divisors. It provides a topological proof of Lagrange inversion and its multiplicative counterpart, expressing CP^n and Theta^n via L_n and M_n polynomials and linking these to the combinatorics of associahedra and permutohedra through a universal formal group law. It further explores divisibility properties of all Chern numbers by the Euler characteristic, introducing numerically extremely divisible varieties and offering numerous examples across dimensions, with connections to Milnor-Hirzebruch type questions. Overall, the work bridges algebraic topology, algebraic geometry, and combinatorics to illuminate how classical inversion formulas arise from characteristic-number data in complex cobordism.
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 formulas in terms of Chern numbers of the smooth theta divisors. We discuss also the general related problem when all Chern numbers of an algebraic variety are divisible by its Euler characteristic.