Equivariant cohomology of (Z2)r-manifolds and syzygies

We consider closed manifolds with (Z(2))(r)-action, which are obtained as intersections of products of spheres of a fixed dimension with certain 'generic' hyperplanes. This class contains the real versions of the 'big polygon spaces' defined and considered by M. Franz (2015). We calculate the equivariant cohomology with F-2-coefficients, which in many examples turns out to be torsion-free but not free and realizes all orders of syzygies, which are in accordance with the restrictions proved by Allday et al. (unpublished). The final results for the real versions are analogous to those for the big polygon spaces in Franz (2015), where (S-1)(r)-actions and rational coefficients are considered, but we consider a wider class of manifolds, and the point of view as well as the method of proof, for which it is essential to consider equivariant cohomology for various related groups, are quite different.