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The $ l\sb 1$-completion of a metric combinatorial $ \infty$-manifold


Author: Katsuro Sakai
Journal: Proc. Amer. Math. Soc. 100 (1987), 574-578
MSC: Primary 57N20; Secondary 54E52, 57Q05
DOI: https://doi.org/10.1090/S0002-9939-1987-0891166-1
MathSciNet review: 891166
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Abstract: Let $ K$ be a simplicial complex. The realization $ \left\vert K \right\vert$ of $ K$ admits the metric

$\displaystyle {d_1}(x,y) = \sum\limits_{\upsilon \in {K^0}} {\left\vert {x(\upsilon ) - y(\upsilon )} \right\vert,} $

where $ x(\upsilon )$ and $ y(\upsilon ),\upsilon \in {K^0}$, are the barycentric coordinates of $ x$ and $ y$ respectively. The completion of the metric space $ (\left\vert K \right\vert,{d_1})$ is called the $ {l_1}$-completion and is denoted by $ {\overline {\vert K\vert} ^{{l_1}}}$. In this paper, we prove that $ {\overline {\vert K\vert} ^{{l_1}}}$ is an $ {l_2}$-manifold if and only if $ K$ is a combinatorial $ \infty $-manifold.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0891166-1
Keywords: Simplicial complex, combinatorial $ \infty $-manifolds, the metric topology, completion, $ l_2^f$-manifold, $ {l_2}$-manifold, $ Q$-manifold, (f.d.) cap set, $ Z$-set
Article copyright: © Copyright 1987 American Mathematical Society

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