Lusternik-Schnirelmann category and Hilbert cube manifolds
For a compact polyhedron P, the Lusternik-Schnirelmann category cat(P) is defined as the smallest integer k such that P can be covered with k subpolyhedra each of which is null homotopic in P. The geometric category gcat(P) is the smallest integer k such that P can be covered with k contractible subpolyhedra. The strong category Cat(P) in the sense of Ganea is the smallest integer k such that \(gcat(K)=k\) for some compact polyhedron K homotopy equivalent to P. In the paper \(Cat(P)=\min_{n\geq 0}gcat(P\times I^ n)\) is proved provided that \(\pi_ 1(P)\) is either free or free abelian. If P is connected then \(cat(P)=smallest\) integer k such that \(P\times Q\) can be covered with \(k+1\) open subsets each of which is homeomorphic to \(Q\times [0,1)\), where Q stands for the Hilbert cube. It is proved that if \(M^ m\) is a compact connected PL m-manifold then \(cat(M^ m)=smallest\) integer k such that \(M^ m\times I^ n\) can be covered with \(k+1\) boundary balls. If \(M^ m\) is a closed connected PL m-manifold then \(cat(M^ m)=smallest\) integer k such that \(M^ m\times S^ n\) can be covered with \(k+1\) PL balls.
scientific article; zbMATH DE number 4054643