  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.

High Energy Physics - Theory