The cohomology of $BPU_n$ in dimensions less than $15$

Abstract

Let $PU_n$ denote the projective unitary group of rank $n$ and $BPU_n$ be its classifying space, for $n>1$. By using the Serre spectral sequence induced by the fibration $BU_n\to BPU_n\to K(\mathbb{Z},3)$, we compute the integral cohomology of $BPU_n$ in dimensions less than $15$ except for $4\mid n$ in dimension $14$.