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Transactions of the American Mathematical Society

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

 
 

 

Approximating topological surfaces in $ 4$-manifolds


Author: Gerard A. Venema
Journal: Trans. Amer. Math. Soc. 265 (1981), 35-45
MSC: Primary 57Q35; Secondary 57N45
DOI: https://doi.org/10.1090/S0002-9947-1981-0607105-3
MathSciNet review: 607105
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Abstract: Let $ {M^2}$ be a compact, connected $ 2$-manifold with $ \partial {M^2} \ne \emptyset $ and let $ h:{M^2} \to {W^4}$ be a topological embedding of $ {M^2}$ into a $ 4$-manifold. The main theorem of this paper asserts that if $ {W^4}$ is a piecewise linear $ 4$-manifold, then $ h$ can be arbitrarily closely approximated by locally flat PL embeddings. It is also shown that if the $ 4$-dimensional annulus conjecture is correct and if $ W$ is a topological $ 4$-manifold, then $ h$ can be arbitrarily closely approximated by locally flat embeddings. These results generalize the author's previous theorems about approximating disks in $ 4$-space.


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DOI: https://doi.org/10.1090/S0002-9947-1981-0607105-3
Keywords: Surface, $ 4$-manifold, topological embedding, piecewise linear approximation, locally flat approximation
Article copyright: © Copyright 1981 American Mathematical Society

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