A Concordance Extension Theorem

Abstract:

Let $p\:E\to B$ be a manifold approximate fibration between closed manifolds, where $\dim (E)\ge 4$, and let $M(p)$ be the mapping cylinder of $p$. In this paper it is shown that if $g\: B\times I\to B\times I$ is any concordance on $B$, then there exists a concordance $G\:M(p)\times I \to M(p)\times I$ such that $G|B\times I=g$ and $G|E\times \{0\}\times I= id_{E\times I}$. As an application, if $N^n$ and $M^{n+j}$ are closed manifolds where $N$ is a locally flat submanifold of $M$ and $n\ge 5$ and $j\ge 1$, then a concordance $g\: N\times I\to N\times I$ extends to a concordance $G\:M\times I\to M\times I$ on $M$ such that $G|N\times I=g$. This uses the fact that under these hypotheses there exists a manifold approximate fibration $p\: E\to N$, where $E$ is a closed $(n+j-1)$-manifold, such that the mapping cylinder $M(p)$ is homeomorphic to a closed neighborhood of $N$ in $M$ by a homeomorphism which is the identity on $N$.