On the Brown-Peterson cohomology of $BP{U}_n$ 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 $BP{U}_n$ to the ordinary cohomology in dimensions $0\le i\le 2p+2$, where $BP{U}_n$ is the classifying space of the projective unitary group $P{U}_n$. Also we show that a family of well understood $p$-torsion cohomology classes $y_{p,k}\in H^{2p^{k+1}+2}(BP{U}_n;Z_{(p)})$ are in the image of the Thom map.

10 pages. Some major changes are made, including some corrections to the main theorems. Comments are welcomed!