De Rham Cohomology of $CP^{1}$ model with Hopf term

We investigate the $CP^{1}$ model possessing the Hopf term which respects the second class constraints and admits the well defined the BRST operator $Q$. Using the operator $Q$, we explicitly construct its de Rham cohomology group by deriving the ensuing quotient group via both the collections of all $Q$-closed and $Q$-exact ghost number $p$-forms. Moreover, we study the $CP^{1}$ model without the Hopf term to evaluate the ensuing effect of the Hopf term on the cohomology group. We find that the Hopf term effects on the de Rham cohomology originate from the Hilbert space modified by this Hopf term.