Algebra of formal vector fields on the line and Buchstaber's conjecture

TL;DR

This paper investigates the cohomology of the Lie algebra of formal vector fields on the line, proving that it is generated by two elements via non-trivial Massey products, advancing understanding of its algebraic structure.

Contribution

It proves that the cohomology H*(L_1) is generated by two elements through non-trivial Massey products, confirming a conjecture about its algebraic generation.

Findings

H*(L_1) is generated by two elements via non-trivial Massey products.

The algebraic structure of H*(L_1) is more complex than trivial cohomology.

Supports Buchstaber's conjecture on the generation of H*(L_1).

Abstract

Let L_1 denotes the Lie algebra of formal vector fields on the line which vanish at the origin together with their first derivatives. Buchstaber and Shokurov have shown that the universal enveloping algebra U(L_1) is isomorphic to the tensor product of the Landweber-Novikov algebra S in complex cobordism theory by reals. The cohomology H*(L_1) has trivial multiplication. Buchstaber conjectured that H*(L_1) is generated with respect to non-trivial Massey products by H^1(L_1). Feigin, Fuchs and Retakh found representation of H*(L_1) by trivial Massey products. In the present article we prove that H*(L_1) is generated with respect to non-trivial Massey products by two elements from H^1(L_1).