Let E be an oriented, smooth and closed m-dimensional manifold with m ≥ 2 and V ⊂ E an oriented, connected, smooth and closed (m-2)-dimensional submanifold which is homologous to zero in E. Let $S^{n-2}\subset Sⁿ$ be the standard inclusion, where Sⁿ is the n-sphere and n ≥ 3. We prove the following extension result: if $h:V\to S^{n-2}$ is a smooth map, then h extends to a smooth map g: E → Sⁿ transverse to $S^{n-2}$ and with $g^{-1}\left(S^{n-2}\right)=V$. Using this result, we give a new and simpler proof of a theorem of Carlos Biasi related to the ambiental bordism...