Let $G$ be a compact , connected and simply connected Lie group and $a\in\mathfrak{g}^\*$ (dual of Lie algebra of Lie group $G$). Then let $O\_a$ be a generic coadjoint orbit then can we say $H^2(O\_a,\mathbb{Z})\cong \mathbb{Z}^n$ for some $n$. Is there any counter example? 



computing second cohomology $H^2(O_a,\mathbb{Z})\cong \mathbb{Z}^n$ of a generic coadjoint orbit
No, there are no counter-examples. Note that a generic coadjoint orbit is $G$-equivariantly diffeomorphic to $G/T$, for a maximal torus $T\subseteq G$. However, $G/T$ (also known as the full flag variety of $G\_{\mathbb{C}}$) has a cell decomposition into even-dimensional cells (the Bruhat decomposition). Therefore, $G/T$ has vanishing integral cohomology in odd degrees, and each even-dimensional integral cohomology group is a free $\mathbb{Z}$-module. In particular, $H^2(G/T;\mathbb{Z})$ is a free $\mathbb{Z}$-module, so your statement is true in general.


