In the talk we introduce the notion of partially framed manifolds, that naturally arise in the study of the characteristic classes of vector bundles over the loop space ΩSU(2) = ΩSP(1). We obtain theorems on the divisibility of the signature of such manifolds as a result of calculations of characteristic classes with values in complex and quaternionic cobordisms. The talk is oriented on a wide audience and therefore includes definitions of basic notions.