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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Obstructions to embedding $ n$-manifolds in $ (2n-1)$-manifolds


Author: J. W. Maxwell
Journal: Trans. Amer. Math. Soc. 180 (1973), 423-435
MSC: Primary 57C35
MathSciNet review: 0321101
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Abstract: Suppose $ f:({M^n},\partial {M^n}) \to ({Q^{2n - 1}},\partial {Q^{2n - 1}})$ is a proper PL map between PL manifolds $ {M^n}$ and $ {Q^{2n - 1}}$ of dimension $ n$ and $ 2n - 1$ respectively, $ M$ compact. J. F. P. Hudson has shown that associated with each such map $ f$ that is an embedding on $ \partial M$ is an element $ \bar \alpha (f)$ in $ {H_1}(M;{Z_2})$ when $ n$ is odd and an element $ \bar \beta (f)$ in $ {H_1}(M;Z)$ when $ n$ is even. These elements are invariant under a homotopy relative to $ \partial M$. We show that, under slight additional assumptions on $ M,Q$ and $ f,f$ is homotopic to an embedding if and only if $ \bar \alpha (f) = 0$ for $ n$ odd and $ \bar \beta (f) = 0$ for $ n$ even. This result is used to give a sufficient condition for extending an embedding $ f:\partial {M^n} \to \partial {B^{2n - 1}}$ ( $ {B^{2n - 1}}$ denotes $ (2n - 1)$-dimensional ball) to an embedding $ F:({M^n},\partial {M^n}) \to ({B^{2n - 1}},\partial {B^{2n - 1}})$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0321101-0
PII: S 0002-9947(1973)0321101-0
Article copyright: © Copyright 1973 American Mathematical Society