in the Lie algebra of the torus T, we have where the sum runs over all connected components F of the set
is the equivariant Euler form of the normal bundle of
The formula allows one to compute the equivariant cohomology ring of the orbifold M (a particular kind of differentiable stack) from the equivariant cohomology of its fixed point components, up to multiplicities and Euler forms.
No analog of such results holds in the non-equivariant cohomology.
One important consequence of the formula is the Duistermaat–Heckman theorem, which states: supposing there is a Hamiltonian circle action (for simplicity) on a compact symplectic manifold M of dimension 2n, where H is Hamiltonian for the circle action, the sum is over points fixed by the circle action and
The localization theorem for equivariant cohomology in non-rational coefficients is discussed in Daniel Quillen's papers.
The localization theorem states that the equivariant cohomology can be recovered, up to torsion elements, from the equivariant cohomology of the fixed point subset.
But there is still a version of the localization theorem for non-abelian actions.