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A $ Q$-manifold local-compactification of a metric combinatorial $ \infty$-manifold

Author: Katsuro Sakai
Journal: Proc. Amer. Math. Soc. 100 (1987), 775-780
MSC: Primary 57N20; Secondary 54E45, 54F40, 57Q15
MathSciNet review: 894453
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Abstract: Let $ K$ be a combinatorial $ \infty $-manifold, that is, a countable simplicial complex such that the star of each vertex is combinatorially equivalent to the countable-infinite full simplicial complex. Then the space $ {\left\vert K \right\vert _m}$ with the metric topology is a manifold modeled on the space $ \sigma $, where $ \sigma $ is the subspace of the Hilbert cube $ Q = {I^\omega }$ which consists of all points having at most finitely many nonzero coordinates. In this paper, we give a local-compactification of $ {\left\vert K \right\vert _m}$ which is a $ [0,1)$-stable $ Q$-manifold containing $ {\left\vert K \right\vert _m}$ as an f.d. cap set.

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  • [An] R. D. Anderson, On sigma-compact subsets of infinite-dimensional spaces, unpublished manuscript.
  • [Ch$ _{1}$] T. A. Chapman, Dense sigma-compact subsets of infinite-dimensional manifolds, Trans. Amer. Math. Soc. 154 (1971), 399-426. MR 0283828 (44:1058)
  • [Ch$ _{2}$] --, On the structure of Hilbert cube manifolds, Compositio Math. 24 (1972), 329-353. MR 0305432 (46:4562)
  • [CDM] D. W. Curtis, T. Dobrowolski, and J. Mogilski, Some applications of the topological characterizations of the sigma-compact spaces $ l_f^2$ and $ \Sigma $, Trans. Amer. Math. Soc. 284 (1984), 837-846. MR 743748 (86i:54035)
  • [Du] J. Dugundji, Locally equiconnected spaces and absolute neighborhood retracts, Fund. Math. 57 (1965), 187-193. MR 0184202 (32:1675)
  • [Hu] S.-T. Hu, Theory of retracts, Wayne State Univ. Press, Detroit, Mich., 1965. MR 0181977 (31:6202)
  • [Ke] O. H. Keller, Die Homoiomorphie der kompakten konvexen Mengen im Hilbertschen Raum, Math. Ann. 105 (1931), 748-758. MR 1512740
  • [Mo] J. Mogilski, Characterizing the topology of infinite-dimensional $ \sigma $-compact manifolds, Proc. Amer. Math. Soc. 92 (1984), 111-118. MR 749902 (85m:57012)
  • [Sa$ _{1}$] K. Sakai, Combinatorial infinite-dimensional manifolds and $ {{\mathbf{R}}^\infty }$-manifolds, Topology Appl. (to appear).
  • [Sa$ _{2}$] -, On topologies of triangulated infinite-dimensional manifolds, J. Math. Soc. Japan 39 (1987) (in press). MR 879930 (88e:57020)
  • [Sa$ _{3}$] -, Simplicial complexes triangulating infinite-dimensional manifolds, preprint.
  • [Sa$ _{4}$] -, Completions of metric simplicial complexes by using $ {l_p}$-norms, preprint.
  • [To] H. Toruńczyk, On CE-images of the Hilbert cube and characterization of $ Q$-manifolds, Fund. Math. 106 (1980), 31-40. MR 585543 (83g:57006)
  • [Wo] R. Y.-T. Wong, Non-compact Hilbert cube manifolds, unpublished manuscript.

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Keywords: Simplicial complex, combinatorial $ \infty $-manifold, metric topology, localcompactification, $ \sigma $-manifold, $ Q$-manifold, $ [0,1)$-stable, f.d. cap set, $ Z$-set, ANR
Article copyright: © Copyright 1987 American Mathematical Society

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