Polynomial Eulerian characteristic of nilmanifolds

V. M. Buchstaber

Funkcionalʹnyj analiz i ego priloženiâ, Tome 58 (2024) no. 1, pp. 22-41

Résumé

The article studies bundle towers $M^{n+1}\to M^{n}\to \dots \to S^1$, $\geqslant 1$, with fiber $S^1$, where $M^n = L^n\!/\Gamma^n$ are compact smooth nilmanifolds and $L^n\thickapprox \mathbb{R}^n$ is a group of polynomial transformations of the line $\mathbb{R}^1$. The focus is on the well-known problem of calculating cohomology rings with rational coefficients of manifolds $M^n$. Using the canonical bigradation in the de Rham complex of manifolds $M^n$, we introduce the concept of polynomial Eulerian characteristic and calculate it for these manifolds.

Détail

Keywords: bigraded de Rham complex, algebra of left invariant differential operators
Mots-clés : polynomial transformations of a line, Gysin exact sequence.