Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1988-0929009-0
MathSciNet review: 929009
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.


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

  • [CH$ _{1}$] T. A. Chapman, Lectures on Hilbert cube manifolds, CBMS Regional Conf. Ser. in Math., no. 28, Amer. Math. Soc., Providence, R. I., 1975. MR 0423357 (54:11336)
  • [CH$ _{2}$] -, Dense sigma-compact subsets of infinite-dimensional manifolds, Trans. Amer. Math. Soc. 154 (1971), 399-426. MR 0283828 (44:1058)
  • [MO$ _{1}$] L. Montejano, Lusternik-Schnirelmann category and Hilbert cube manifolds, Preprint.
  • [MO$ _{2}$] -, A quick proof of Singhof's $ \operatorname{cat} \left( {M \times {S^1}} \right) = \operatorname{cat} \left( M \right) + 1$ theorem, Manuscripta Math. 42 (1983), 49-52. MR 693418 (85a:55002)
  • [JA] I. M. James, On category, in the sense of Lusternik-Schnirelmann, Topology 19 (1978), 331-349. MR 516214 (80i:55001)
  • [WE] J. E. West, Infinite products which are Hilbert cubes, Trans. Amer. Math. Soc. 150 (1970), 1-25. MR 0266147 (42:1055)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57N20, 55M30, 58B05

Retrieve articles in all journals with MSC: 57N20, 55M30, 58B05


Additional Information

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

American Mathematical Society