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