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

TL;DR

This paper investigates the image of the Thom map from Brown-Peterson cohomology to ordinary cohomology for the classifying space of the projective unitary group, revealing specific torsion classes in lower dimensions.

Contribution

It determines the image of the Thom map in certain dimensions and identifies particular p-torsion classes as being in this image.

Findings

Thom map image characterized in dimensions 0 to 2p+2.

Identification of p-torsion classes y_{p,k} in the image.

Results applicable for odd primes p.

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.