Connections with prescribed first Pontrjagin form

Abstract:

Let $P$ be a principal $O(n)$ bundle over a $C^\infty$ manifold $M$ of dimension $m$. If $n\geq 5m+4+4\binom {m+1}{4}$, then we prove that every differential 4-form representing the first Pontrjagin class of $P$ is the Pontrjagin form of some connection on $P$.