The 𝑙₁-completion of a metric combinatorial ∞-manifold

Sakai, Katsuro

doi:10.1090/S0002-9939-1987-0891166-1

The $l_ 1$-completion of a metric combinatorial $\infty$-manifold
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Proc. Amer. Math. Soc. 100 (1987), 574-578 Request permission

Abstract:

Let $K$ be a simplicial complex. The realization $\left | K \right |$ of $K$ admits the metric \[ {d_1}(x,y) = \sum \limits _{\upsilon \in {K^0}} {\left | {x(\upsilon ) - y(\upsilon )} \right |,} \] 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 $(\left | K \right |,{d_1})$ is called the ${l_1}$-completion and is denoted by ${\overline {|K|} ^{{l_1}}}$. In this paper, we prove that ${\overline {|K|} ^{{l_1}}}$ is an ${l_2}$-manifold if and only if $K$ is a combinatorial $\infty$-manifold.

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