A Hilbert cube L-S category

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 $P$.