  We consider closed manifolds, which occur as intersections of products of
spheres of the same dimension with certain hyperplanes. Among those are the so
called (big) polygon- and chain spaces. The equivariant cohomology with respect
to natural actions of 2-tori is calculated and related to the notion of syzygy.
It turns out that the equivariant cohomology module of these manifolds is often
torsion free, but not free over the cohomology of the group. Coefficients are
taken in the field with two elements.
