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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Covering isotopies of $ M\sp{n-1}$ in $ N\sp{n}$


Author: Perrin Wright
Journal: Proc. Amer. Math. Soc. 29 (1971), 591-598
MSC: Primary 57.01
MathSciNet review: 0281215
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Abstract: We show that a continuous family of locally flat separating embeddings of an $ (n - 1)$-manifold $ {M^{n - 1}}$ into an n-manifold $ {N^n}$, where the family is parametrized by a locally compact finite-dimensional metric space B, can be covered locally and sometimes globally by a continuous family of homeomorphisms of $ {N^n}$ onto itself, provided $ n \ne 4$. Furthermore, the covering family can be chosen to extend a preassigned covering family corresponding to a compact connected subset of B. We derive a stronger result for embeddings of $ {S^{n - 1}}$ in $ {S^n}$, and show that the natural map from the space of orientation preserving homeomorphisms of $ {S^n}$ to the space of locally flat embeddings of $ {S^{n - 1}}$ into $ {S^n},n \ne 4$, is a Serre fibration and a weak homotopy equivalence.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0281215-4
PII: S 0002-9939(1971)0281215-4
Keywords: Manifold, locally flat embedding, locally contractible, completely regular mapping, Serre fibration, weak homotopy equivalence
Article copyright: © Copyright 1971 American Mathematical Society