3 lectures on foliations and laminations on 3-manifolds (Q2725683)

The paper under review is a survey of how (codimension-1) foliations and laminations are used to derive topological information about orientable \(3\)-manifolds.NEWLINENEWLINENEWLINEIn section 1, important results on foliations are presented. A foliation of a \(3\)-manifold \(M\) is a decomposition of \(M\) into (possibly non-compact) surfaces called leaves such that \(M\) is covered by a collection of charts of the form \({\mathbf R}^2 \times {\mathbf R}\) where the leaves pass through every chart in slices of the form \({\mathbf R}^2 \times x\), \(x \in {\mathbf R}\). We recall some of the results. (1.8) If \(M\) is closed and admits a foliation with an \(S^2\) leaf, then \(M = S^2 \times S^1\) or \(P^3 \sharp P^3\). (1.10) If \(M (\neq S^2 \times S^1)\) admits a Reebless foliation, then the universal cover \(\widetilde{M}\) is \({\mathbf R}^3\). (1.12) A \(C^2\)-foliation without holonomy is defined by a closed nonsingular \(1\)-form. (1.14) If \(M\) is closed and has a \(C^0\)-foliation by planes, then \(M = T^3\). (1.15) If \(M\) has a foliation defined by a closed 1-form, then \(M\) fibres over \(S^1\).NEWLINENEWLINENEWLINESection 2 is on taut foliations on \(3\)-manifolds. A foliation \({\mathcal F}\) is called taut if there exists a vector field transverse to the leaves and has a closed transversal which intersects each leaf of \({\mathcal F}\). (2.10) If \({\mathcal F}\) is a \(C^2\) transversely orientable foliation on a compact \(3\)-manifold, then the following five conditions (a)--(e) are equivalent. (a) \({\mathcal F}\) is taut. (b) \({\mathcal F}\) has no dead end components (in particular, is Reebless). (c) The leaves of \({\mathcal F}\) are homologically norm minimizing. (d) There exists a Riemannian metric such that the leaves of \({\mathcal F}\) are homologically area minimizing. (e) There exists a volume preserving flow transverse to \({\mathcal F}\). (2.11) Suppose that \(M\) is compact and irreducible and that \(\partial M\) is empty or a union of tori. If \(S\) is a norm minimizing surface, then there is a taut and finite depth foliation \({\mathcal F}\) of \(M\) such that \(S\) is a leaf of \({\mathcal F}\) and that \({\mathcal F}|_{\partial M}\) is a Reebless foliation. An argument of construction of taut foliation proves (2.18) the Property R and Poenaru Conjectures (on \(0\)-surgeries on knots yielding \(S^2 \times S^1\) summands) are true.NEWLINENEWLINENEWLINESection 3 is on essential laminations, which is a simultaneous generalization of both incompressible surfaces and taut foliations. A lamination on a \(3\)-manifold \(M\) is a closed subset \(\lambda\) of \(M\) such that \(M\) is covered by a collection of charts of the form \({\mathbf R}^2 \times {\mathbf R}\) with \(\lambda \cap ({\mathbf R}^2 \times {\mathbf R}) = {\mathbf R}^2 \times K\), where \(K\) is a closed subset of \({\mathbf R}\). (3.5) If \(M\) contains an essential lamination, then the universal cover \(\widetilde{M}\) is \({\mathbf R}^3\). (3.16) If \(M\) is atoroidal and contains a genuine lamination, then (like a Haken manifold) (a) \(\pi_1 (M)\) is negatively curved, (b) \(|\pi_0 (\text{Diff}(M))|< \infty\) and (c) homotopic homeomorphisms of \(M\) are isotopic if some non I-bundle complementary region is a solid torus. (3.25) If \(M\) contains a genuine lamination, then Homeo\((M)\)/Homeo\({}_0 (M)\) is finite.NEWLINENEWLINENEWLINEMany other technical but important results are presented. In Definition 3.9, the notion of loosesse laminations is introduced. It is weaker than that of essential laminations, and the author expects that it is perhaps a more plentiful object. Many problems and conjectures are presented. On Problem 3.7.3, the preprint [An algorithm to detect laminar 3-manifolds] by \textit{L. Agol} and \textit{T. Li} gives an affirmative answer.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00033].