The Cohomology of Projective Unitary Groups

The projective unitary group \(\mathrm{PU}(n)\) is the quotient of the unitary group \(\mathrm{U}(n)\) by its center \(S^1=\{e^{i\theta}I_n \colon\, \theta\in [0,2\pi]\}\) , where \(I_n\) is the identity matrix. Combining the Serre spectral sequence of the fibration \(\mathrm{PU}(n)\to \mathrm{PU}(n)/T\) with the Gysin sequence of the circle bundle \(\mathrm{U}(n)\to \mathrm{PU}(n)\) , we compute the integral cohomology ring of \(\mathrm{PU}(n)\) using explicitly constructed generators, where \(T\) is a maximal torus of \(\mathrm{PU}(n)\) .