Codimension one minimal foliations and the higher homotopy groups of leaves
Let \({\mathcal F}\) be a codimension one foliation on an aspherical manifold \(M\), where \({\mathcal F}\) has no vanishing cycles. In this paper the author shows the asphericity of foliations under some additional assumptions. A topological space \(X\) is said to be \(k\)-aspherical if the universal covering of \(X\) is \(k\)-connected. Theorem 1: ``Let \({\mathcal F}\) be a codimension one foliation without vanishing cycles of a \(k\)-aspherical manifold \(M\), where \(k\geq 3\). If there is a dense leaf of \({\mathcal F}\) which is \((k-1)\)-aspherical, then each leaf of \({\mathcal F}\) is \((k-1)\)-aspherical. In particular, for an aspherical manifold \(M\) if there is an aspherical dense leaf of \({\mathcal F}\), then each leaf of \({\mathcal F}\) is aspherical.'' Theorem 2: ``Let \({\mathcal F}\) be a codimension one minimal foliation without vanishing cycles of a manifold \(M\). Suppose that \(M\) is \((k+1)\)-aspherical. If there is a leaf of \({\mathcal F}\) which is exactly \((k-1)\)-aspherical, then each leaf of \({\mathcal F}\) is exactly \((k-1)\)-aspherical.'' The author gives an example of a minimal foliation of an exactly \((k-1)\)-aspherical manifold \(M\) with exactly \((k-2)\)-aspherical leaves and exactly \((k-1)\)-aspherical leaves.
scientific article; zbMATH DE number 5563771