L'anneau de cohomologie de toutes les vari\'et\'es de Seifert

TL;DR

This paper computes the cohomology ring with coefficients in rac{1}{p} of all Seifert manifolds, both orientable and non-orientable, using a simplicial decomposition and detailed choices for cup-product calculations.

Contribution

It provides a method to explicitly determine the cohomology ring of Seifert manifolds from a simplicial decomposition, including the necessary choices for cup-product computations.

Findings

Cohomology ring structure derived for all Seifert manifolds.

Explicit procedure for lifting from cellular to simplicial complexes.

Handling of non-orientable cases in cohomology computations.

Abstract

The cohomology ring with coefficients in $\Z_p$, where $p$ is a prime integer, of a Seifert manifold $M$, orientable or not orientable is obtained from a simplicial decomposition of $M$. Many choices must be made before applying Alexander-Whitney formula to get the cup-products. The most difficult choices are those of liftings from the cellular complex to the simplicial complex when we add the condition to be cocycles.