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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Obstructions to embedding $n$-manifolds in $(2n-1)$-manifolds
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by J. W. Maxwell PDF
Trans. Amer. Math. Soc. 180 (1973), 423-435 Request permission

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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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 180 (1973), 423-435
  • MSC: Primary 57C35
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0321101-0
  • MathSciNet review: 0321101