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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A Hilbert cube L-S category


Author: Raymond Y. Wong
Journal: Proc. Amer. Math. Soc. 102 (1988), 720-722
MSC: Primary 57N20; Secondary 55M30, 58B05
MathSciNet review: 929009
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Abstract: Let $ M$ be a compact connected Hilbert cube manifold ($ Q$-manifold). Define $ {C_z}\left( M \right)$ to be the smallest integer $ k$ such that $ M$ can be covered with $ k$ open subsets each of which is homeomorphic to $ Q \times \left[ {0,1} \right)$. Recently L. Montejano proved that, for every compact connected polyhedron $ P,{C_z}\left( {P \times Q} \right) = \operatorname{cat}\left( P \right) + 1$, where $ \operatorname{cat} \left( P \right)$ is the Lusternik-Schnirelmann category of $ P$. Using a different approach, we prove a noncompact analog of the above theorem by showing that $ {C_z}\left( {P \times Q \times \left[ {0,1} \right)} \right) = \operatorname{cat}\left( P \right)$ for every compact connected polyhedron $ P$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0929009-0
PII: S 0002-9939(1988)0929009-0
Keywords: Lusternik-Schnirelmann category, Hilbert cube manifold, polyhedron, $ {\mathbf{Z}}$-sets
Article copyright: © Copyright 1988 American Mathematical Society