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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Infinite products which are Hilbert cubes


Author: James E. West
Journal: Trans. Amer. Math. Soc. 150 (1970), 1-25
MSC: Primary 54.25
DOI: https://doi.org/10.1090/S0002-9947-1970-0266147-3
MathSciNet review: 0266147
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Abstract: Let $Q$ denote the Hilbert cube. It is shown that if $P$ and $P’$ are compact polyhedra of the same simple homotopy type then $P \times Q$ and $P’ \times Q$ are homeomorphic. A consequence of this result is that the Cartesian product of a countable, locally finite simplicial complex with a separable, infinite-dimensional Fréchet space is a manifold modelled on the Fréchet space. It is also proved that a countably infinite product of nondegenerate spaces is a Hilbert cube provided that the product of each of the spaces with the Hilbert cube is a Hilbert cube. Together with the first result, this establishes that a countably infinite product of nondegenerate, compact, contractible polyhedra is a Hilbert cube. In addition, a proof is given of the (previously unpublished) theorem of R. D. Anderson that a countably infinite product of nondegenerate dendra is a Hilbert cube.


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Keywords: Hilbert cube, infinite product, polyhedron, simple homotopy type, local homotopy negligibility, Property <IMG WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$Z$">
Article copyright: © Copyright 1970 American Mathematical Society