Step 2. Gluing identity. Consider the vector bundle U d := π * V d → M d (X), restricted to the fixed point components F r . A point in (C, f ) in F r is a pair (C 1 , f 1 , x 1 )×(C 2 , f 2 , x 2 ) of 1-pointed stable maps glued together at the marked points, ie. f 1 (x 1 ) = f 2 (x 2 ). From this, we get an exact sequence of bundles on F r :
0 → i * r U d → U ′ r ⊕ U ′ d−r → e * V → 0.
Here i * r U d is the restriction to F r of the bundle U d → M d (X). And U ′ r is the pullback of the bundle U r → M 0,1 (d, X) induced by V , and similarly for U ′ d−r . Taking the multiplicative characteristic class b, we get the identity on F r :
e * b(V )b(i * r U d ) = b(U ′ r )b(U ′ d−r ).
This is what we call the gluing identity. This may be translated to a similar quadratic identity, via Step 1, for Q d in the equivariant cohomology groups H * S 1 (W d ). The new identity is called the Euler **data** identity.