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 $$\mathrm{PU}(n)$$ .