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Each $\textbf {R}^{\infty }$-manifold has a unique piecewise linear $\textbf {R}^{\infty }$-structure


Author: Katsuro Sakai
Journal: Proc. Amer. Math. Soc. 90 (1984), 616-618
MSC: Primary 57N20; Secondary 57Q25, 58B05
DOI: https://doi.org/10.1090/S0002-9939-1984-0733416-5
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.


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

  • Richard E. Heisey, Embedding piecewise linear ${\bf R}^{\infty }$-manifolds into ${\bf R}^{\infty }$, Topology Proc. 6 (1981), no. 2, 317–328 (1982). MR 672463
  • Richard E. Heisey, Manifolds modelled on the direct limit of lines, Pacific J. Math. 102 (1982), no. 1, 47–54. MR 682043
  • 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
  • K. Sakai, On ${{\mathbf {R}}^\infty }$-manifolds and ${Q^\infty }$-manifolds, Topology Appl. (to appear).

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Keywords: Direct limit, <!– MATH ${{\mathbf {R}}^\infty }$ –> <IMG WIDTH="40" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="${{\mathbf {R}}^\infty }$">-manifold, p.l. <!– MATH ${{\mathbf {R}}^\infty }$ –> <IMG WIDTH="40" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img4.gif" ALT="${{\mathbf {R}}^\infty }$">-structure, p.l. <!– MATH ${{\mathbf {R}}^\infty }$ –> <IMG WIDTH="40" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img15.gif" ALT="${{\mathbf {R}}^\infty }$">-manifold, <!– MATH ${{\mathbf {R}}^\infty }$ –> <IMG WIDTH="40" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${{\mathbf {R}}^\infty }$">-p.l. map, <!– MATH ${{\mathbf {R}}^\infty }$ –> <IMG WIDTH="40" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="${{\mathbf {R}}^\infty }$">-p.l. isomorphism, polyhedron
Article copyright: © Copyright 1984 American Mathematical Society