I am currently studying cohomology groups of complex algebraic varieties, spending a lot of time on computing Betti numbers. I came across several manifolds which have a discrete symmetry group (like e.g. $\mathbb{Z}_2$). So my question: For a complex algebraic variety $\mathcal{M}$, is it possible to relate the Betti numbers of $\mathcal{M}$ to those of $\mathcal{M}\times \mathbb{Z}_n$? I know that in general for product spaces, the Künneth formula can be used to get the Betti-numbers. But is this also true if a finite group is multiplied with a smooth manifold?
Thanks in advance, Max