Virtual fundamental class

The moduli space of maps ${\overline{\mathcal{M}}}_{g,n}(X,\beta )$ is highly singular, and yet it has a homology class which behaves much like a fundamental class, and is hence called the virtual fundamental class $[{\overline{\mathcal{M}}}_{g,n}(X,\beta ){]}^{\mathrm{vir}}$ defined in terms of the deformation theory of stable maps to $X$. The Gromov-Witten invariants of $X$ are defined by integrating various geometric cohomology classes over this class.