On the Extension of Certain Maps with Values in Spheres
Volume 56 / 2008
Abstract
Let $E$ be an oriented, smooth and closed $m$-dimensional manifold with $m \ge 2$ and $V \subset E$ an oriented, connected, smooth and closed $(m-2)$-dimensional submanifold which is homologous to zero in $E$. Let $S^{n-2} \subset S^n$ be the standard inclusion, where $S^n$ is the $n$-sphere and $n \ge 3$. We prove the following extension result: if $h:V \to S^{n-2}$ is a smooth map, then $h$ extends to a smooth map $g:E \to S^n$ transverse to $S^{n-2}$ and with $g^{-1}(S^{n-2})=V$. Using this result, we give a new and simpler proof of a theorem of Carlos Biasi related to the \it ambiental bordism \rm question, which asks whether, given a smooth closed $n$-dimensional manifold $E$ and a smooth closed $m$-dimensional submanifold $V \subset E$, one can find a compact smooth $(m+1)$-dimensional submanifold $W \subset E$ such that the boundary of $W$ is $V$.
