H^2\big(\ov\M_{g,S}(X,\be;J)\big)$$ the first chern class of the universal cotangent line bundle for the $j$-th marked point. The space $\ov\M_{g,S}(X,\be;J)$ carries a natural VFC, which is independent of $J$ and will be denoted by $[\ov\M_{g,S}(X,\be)]^{vir}$. If the (real) dimension of $X$ is $2n$, then \^[vir]{} =\_[g,S]{}(X,) 2(+(n-3)(1-g)+|S|).If $J$ is regular[^3], then $\ov\M_{0,S}(X,\be;J)$ is a topological manifold with a preferred choice of orientation and $$\big[\ov\M_{0,S}(X,\be)\big]^{vir}=\big[\ov\M_{0,S}(X,\be;J)\big].$$\
If $a_j\!\in\!\bar\Z^+$ and $\ka_j\!\in\!H_*(X;\Z)$ for each $j\!\in\!S$, let ((\_[a\_j]{}\_j)\_[jS]{})\_[g,]{}\^X \_[jS]{} (\_j\^[a\_j]{}\_j\^\*(\_X\_j)),\^[vir]{},where $\PD_X\ka_j\!\in\!H^*(X;\Z)$ is the Poincare dual of $\ka_j$ in $X$.[^4] In order to avoid any sign ambiguities, we define the number in \_ref[GWdfn\_e]{} to be $0$ if the dimension of $\ka_j$ is odd for some $j$. By \_ref[virdim\_e]{}, this number is zero unless \_[jS]{}(2a\_j+2n-\_j)=\_[g,S]{}(X,).The number \_ref[GWdfn\_e]{} can be expressed as an integral on a “smaller" moduli space as follows. Choose cobordism representatives $f_j\!:M_j\!\lra\!X$ for $\ka_j$, with $j\!\in\!S$.[^5] Let \_[g,]{}(X,;J)={(\[u\],(w\_j)\_[jS]{}) \_[g,S]{}(X,;J)\_[jS]{}M\_j: \_j(\[u\])=f\_j(w\_j) jS}.The space $\ov\M_{g,\f}(X,\be;J)$ of constrained stable maps also carries a virtual fundamental class and $$\big((\tau_{a_j}\ka_j)_{j\in S}\big)_{g,\be}^X =
\bigg\lan\prod_{j\in S}\psi_j^{a_j},\big[\ov\M_{g,\f}(X,\be;J)\big]^{vir}\bigg\ran.$$ The subject of this section is a reduction of this GW-invariant of $X$ to a combination of GW-invariants for its submanifolds.

\[propinter\_dfn\] Let $Y$ be a submanifold of $X$. A smooth map $f\!:M\!\lra\!X$ intersects $Y$ if $f^{-1}(Y)\!\subset\!M$ 