Step 3. Linking theorem. The construction above is functorial, so that if X comes equipped with a torus T action, then the entire construction becomes G = S 1 × T equivariant and not just S 1 equivariant. In particular, the Euler **data** identity is an identity of G-equivariant classes on W d . Our problem is to first compute the G-equivariant classes Q d on W d satisfying the Euler **data** identity, and with the property that A d ∼ α −2 . Note that the restrictions Q d | p to the T fixed points p in X 0 ⊂ W d are polynomial functions on the Lie algebra of G. Suppose that X is a balloon manifold. Then it can be shown that (with a nondegeneracy assumption on e G (X 0 /W d )) the classes Q d are uniquely determined by the values of the Q d | p , when α is some scalar multiple of a weight on the tangent space T p X. These values of Q d | p can be computed explicitly by exploiting the structure of a balloon manifold.