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Each $ {\bf R}\sp{\infty }$-manifold has a unique piecewise linear $ {\bf R}\sp{\infty }$-structure

Author: Katsuro Sakai
Journal: Proc. Amer. Math. Soc. 90 (1984), 616-618
MSC: Primary 57N20; Secondary 57Q25, 58B05
MathSciNet review: 733416
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Abstract: R. E. Heisey introduced piecewise linear $ {{\mathbf{R}}^\infty }$-structures and defined piecewise linear $ {{\mathbf{R}}^\infty }$-manifolds. In this paper we show that two piecewise linear $ {{\mathbf{R}}^\infty }$-manifolds are isomorphic if they have the same homotopy type. From the Open Embedding Theorem for (topological) $ {{\mathbf{R}}^\infty }$-manifolds and this result, we have the title.

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  • [1] Richard E. Heisey, Embedding piecewise linear 𝑅^{∞}-manifolds into 𝑅^{∞}, Topology Proc. 6 (1981), no. 2, 317–328 (1982). MR 672463
  • [2] Richard E. Heisey, Manifolds modelled on the direct limit of lines, Pacific J. Math. 102 (1982), no. 1, 47–54. MR 682043
  • [3] C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69. MR 0350744
  • [4] K. Sakai, On $ {{\mathbf{R}}^\infty }$-manifolds and $ {Q^\infty }$-manifolds, Topology Appl. (to appear).

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Keywords: Direct limit, $ {{\mathbf{R}}^\infty }$-manifold, p.l. $ {{\mathbf{R}}^\infty }$-structure, p.l. $ {{\mathbf{R}}^\infty }$-manifold, $ {{\mathbf{R}}^\infty }$-p.l. map, $ {{\mathbf{R}}^\infty }$-p.l. isomorphism, polyhedron
Article copyright: © Copyright 1984 American Mathematical Society