  This article is devoted to the investigation of wrap groups of connected
fiber bundles over the fields of real $\bf R$, complex $\bf C$ numbers, the
quaternion skew field $\bf H$ and the octonion algebra $\bf O$. Cohomologies of
wrap groups and their structure are investigated. Sheaves of wrap groups are
constructed and studied. Moreover, twisted cohomologies and sheaves over
quaternions and octonions are investigated as well.
