The $l_ 1$-completion of a metric combinatorial $\infty$-manifold
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- by Katsuro Sakai PDF
- 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.References
-
R. D. Anderson, On sigma-compact subsets of infinite-dimensional spaces, unpublished manuscript.
- T. A. Chapman, Dense sigma-compact subsets of infinite-dimensional manifolds, Trans. Amer. Math. Soc. 154 (1971), 399β426. MR 283828, DOI 10.1090/S0002-9947-1971-0283828-7
- T. A. Chapman, Lectures on Hilbert cube manifolds, Regional Conference Series in Mathematics, No. 28, American Mathematical Society, Providence, R.I., 1976. Expository lectures from the CBMS Regional Conference held at Guilford College, October 11-15, 1975. MR 0423357, DOI 10.1090/cbms/028
- David W. Henderson, Infinite-dimensional manifolds are open subsets of Hilbert space, Bull. Amer. Math. Soc. 75 (1969), 759β762. MR 247634, DOI 10.1090/S0002-9904-1969-12276-7
- Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
- Katsuro Sakai, Combinatorial infinite-dimensional manifolds and $\textbf {R}^\infty$-manifolds, Topology Appl. 26 (1987), no.Β 1, 43β64. MR 893803, DOI 10.1016/0166-8641(87)90025-3
- Katsuro Sakai, On topologies of triangulated infinite-dimensional manifolds, J. Math. Soc. Japan 39 (1987), no.Β 2, 287β300. MR 879930, DOI 10.2969/jmsj/03920287 β, Simplicial complexes triangulating infinite-dimensional manifolds, preprint. β, Completions of metric simplicial complexes by using ${l_p}$-norms, Topology Proc. 11 (1986). β, A $Q$-manifold local-compactification of a metric combinatorial $\infty$-manifold, Proc. Amer. Math. Soc. 100 (1987), in press.
- H. ToruΕczyk, On $\textrm {CE}$-images of the Hilbert cube and characterization of $Q$-manifolds, Fund. Math. 106 (1980), no.Β 1, 31β40. MR 585543, DOI 10.4064/fm-106-1-31-40
Additional Information
- © Copyright 1987 American Mathematical Society
- 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