Cohomology ring of a closed oriented surface ($H^{\ast }({\Sigma }_g;\mathbb{Z})$)

Choose degree-one classes ${\alpha }_i,{\beta }_i$ dual to a symplectic basis of one-cycles and let $\omega$ generate $H^2({\Sigma }_g;\mathbb{Z})$. The only nonzero products of positive-degree basis elements are