The proper L-S category of Whitehead manifolds
The main result of the paper states that the proper LS category of an eventually end-irreducible, \(\mathbb{R}^2\)-irreducible Whitehead 3-manifold is 4. Recall that a Whitehead manifold is a contractible open 3-manifold such that any embedded sphere bounds a ball. The manifold is \(\mathbb{R}^2\)-irreducible if any properly embedded plane \(\mathbb{R}^2\subset M\) bounds a half-space \(\mathbb{R}^2\times\mathbb{R}_+\subset M\). For a Whitehead 3-manifold, this is equivalent to \(\varprojlim\,\text{pro\,}\pi_1(X\alpha)= 0\) for any ray \(\alpha: \mathbb{R}_+\to M\). The manifold \(M\) is eventually end-irreducible if there exists an exhausting sequence \(\{M_n\}\) in \(M\) consisting of compact connected submanifolds such that there is \(n_0\) for which each surface \(\partial M\) is incompressible in \(\overline{M- M_{n_0}}\) if \(n> n_0\). This is equivalent to saying that pro-\(\pi_1(X,\alpha)\) is isomorphic to a tower whose bounding homomorphisms are injective.
scientific article; zbMATH DE number 5012977
scientific article