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Transactions of the American Mathematical Society

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

 

 

Locally flat imbeddings of topological manifolds in codimension three.


Author: Glenn P. Weller
Journal: Trans. Amer. Math. Soc. 157 (1971), 161-178
MSC: Primary 57.01
DOI: https://doi.org/10.1090/S0002-9947-1971-0276981-2
MathSciNet review: 0276981
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Abstract: This paper presents an imbedding theorem for one topological manifold $ {M^n}$ in another topological manifold $ {Q^q}$, provided that the codimension $ (q - n)$ is at least three. The result holds even if the manifolds are of the recently discovered non-piecewise-linear type. Denote the boundaries of $ M$ and $ Q$ by $ \dot M$ and $ \dot Q$ respectively. Suppose that $ M$ is $ 2n - q$ connected and $ Q$ is $ 2n - q + 1$ connected. It is then proved that any map $ f:(M,\dot M) \to (Q,\dot Q)$ such that $ f\vert\dot M$ is a locally flat imbedding is homotopic relative to $ \dot M$ to a proper locally flat imbedding $ g:M \to Q$. It is also shown that if $ M$ is closed and $ 2n - q + 1$ connected and $ Q$ is $ 2n - q + 2$ connected, then any two homotopic locally flat imbeddings are locally flatly concordant.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0276981-2
Keywords: Topological manifold, locally flat imbedding, codimension three, concordant imbeddings, engulfing, piecewise-linear manifold
Article copyright: © Copyright 1971 American Mathematical Society