The cohomology ring of a class of Seifert manifolds

The cohomology groups of the Seifert manifolds are well known. In this article a method is given to compute the cup products in the cohomology ring of any orientable Seifert manifold whose associated orbit surface is S2 , and for any coefficients. In particular the Z/2 cohomology ring is completely determined. This is applied to determine the existence of degree 1 maps from the Seifert manifold to RP3 , and to the Lusternik–Schnirelmann category.