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 $S^1=\{e^{i\theta }I_n: \theta \in [0,2\pi ]\}$, 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 $\mathrm {PU}(n)$.