On embedding of total spaces of fiber bundles over a circle with the Cantor set as a fiber in two-dimensional manifolds

The problem of an embedding of total spaces of fiber bundles over a circle with the Cantor set as a fiber (Pontryagin bundles) in two-dimensional manifolds is investigated. The sufficient condition is obtained for nonexistence of a two-dimensional manifold $M^{2}$ and inclusion map $\Phi\colon N\to M^2$ for total space $N$ of the Pontryagin bundle $\xi=(N,p,S^1)$. We also construct the extensive class of spaces satisfying the condition.