On the Brown-Peterson cohomology of $BPU_n$ in lower dimensions and the Thom map

Abstract

For an odd prime $p$, we study the image of the Thom map from Brown-Peterson cohomology of $BPU_n$ to the ordinary cohomology in dimensions $0\leq i\leq 2p+2$, where $BPU_n$ is the classifying space of the projective unitary group $PU_n$. Also we show that a family of well understood $p$-torsion cohomology classes $y_{p,k}\in H^{2p^{k+1}+2}(BPU_n;Z_{(p)})$ are in the image of the Thom map.