Theorem
Let \(f:X\to Y\) be a \(T\)-equivariant morphism of non-singular schemes with actions of a torus \(T\). For each connected component \(P\) of \(X^T\), \(f(P)\) is contained in a unique component \(Q\) of \(Y^T\); denote by \(f_P: P\to Q\) the restriction of \(f\) to \(P\). We also write \(u|_P\) for the image of class \(u\in H^*_T(X)\) under the pullback of \(\iota:P\hookrightarrow X\). Then for any \(u\in H^*_T(X)\) we have \[f_*(U)|_Q = c^{T}_{\text{top}}(N_{Q/Y}) \cdot \sum_{P \text{ with }f(P) \subseteq Q}(f_P)_*\left(\frac{u|_P}{c^T_{\text{top}}(N_{P/X})}\right)\]
In the case that \(Y = \pt\), we get an integration formula for \(\rho:X\to \pt\): \[\rho_*(u) = \sum_{P\subseteq X^T} (\rho_P)_*\left(\frac{u|_P}{c^T_{\text{top}}(N_{P/X})}\right),\] where \((\rho_P)_*:H^*_T(P)\to \Lambda\) is the Gysin pushforward, i.e. integration over the connected component \(P\subseteq X^T\).