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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A Concordance Extension Theorem

Author: Joel L. Jones
Journal: Trans. Amer. Math. Soc. 348 (1996), 205-218
MSC (1991): Primary 57N37; Secondary 55R65, 57N70
MathSciNet review: 1303122
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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 $g\colon B\times I\to B\times I$ 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 $g\colon N\times I\to N\times I$ 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 $p\colon E\to N$, 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 $N$.

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Additional Information

Joel L. Jones
Affiliation: Department of Mathematics, Presbyterian College, Clinton, South Carolina 29325

Keywords: Concordance, manifold approximate fibration, mapping cylinder
Received by editor(s): October 31, 1994
Article copyright: © Copyright 1996 American Mathematical Society