On the integral cohomology of quotients of manifolds by cyclic groups

We propose new tools based on basic lattice theory to calculate the integral cohomology of the quotient of a manifold by an automorphism group of prime order. As examples of applications, we provide the Beauville–Bogomolov forms of some irreducible symplectic orbifolds; we also show a new expression for a basis of the integral cohomology of a Hilbert scheme of two points on a surface.