An application of this principle is the Borel construction from algebraic topology. Assuming that G is compact, let EG denote the universal bundle, which we can assume to be a manifold since G is compact, and let G act on EG \times M diagonally. The action is free since it is so on the first factor and is proper since G is compact; thus, one can form the quotient manifold M_G = (EG \times M)/G and define the equivariant cohomology of M as
