On the Cohomology of the Classifying Spaces of Projective Unitary Groups

Let $\mathbf{B}P{U}_n$ be the classifying space of $P{U}_n$, the projective unitary group of order $n$, for $n>1$. We use the Serre spectral sequence associated to a fiber sequence $\mathbf{B}{U}_n\to \mathbf{B}P{U}_n\to K(\mathbb{Z},3)$ to determine the ring structure of $H^{\ast }(\mathbf{B}P{U}_n;\mathbb{Z})$ up to degree $10$, as well as a family of distinguished elements of $H^{2p+2}(\mathbf{B}P{U}_n;\mathbb{Z})$, for each prime divisor $p$ of $n$. We also study the primitive elements of $H^{\ast }(\mathbf{B}{U}_n;\mathbb{Z})$ as a comodule over $H^{\ast }(K(\mathbb{Z},2);\mathbb{Z})$, where the comodule structure is given by an action of $K(\mathbb{Z},2)\simeq \mathbf{B}{S}^1$ on $B{U}_n$ corresponding to the action of taking the tensor product of a complex line bundle and an $n$ dimensional complex vector bundle.