1983/01/01 by David Gabai · 902 citations
Mathematics · #Geometric and Algebraic Topology #Geometric Analysis and Curvature Flows #Advanced Operator Algebra Research #Mathematics #Topology (electrical circuits) #Pure mathematics #Combinatorics
paper · pdf · doi:10.4310/jdg/1214437784
published in Journal of Differential Geometry 18(3) (Lehigh University)
openalex publication_date 1983/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Corollary 6.5.A nontήvial link Lin S 3 is nonsplit if and only ifL is the set of cores of Reeb components of some foliation ΦofS 3 .The => direction follows from Theorem 5.5.Novikov [21] proved the converse in 1965.We therefore answer the so-called "Reeb placement problem" of Laudenbach and Roussarie [16] who asked which links could be realized as cores of Reeb components of foliations of S 3 .The holonomy of our foliations along the toral leaves is in general C°.The C 00 problem is open although it can be solved for the alternating knots, fibred knots, many other knots, and certain "sums" of such knots using the constructions [6]-[8]Corollary 6.7.Let R t be a Seifert surface for the oriented link L, C S 3 for i -1,2, and R be any Murasugi sum (or generalized plumbing) of R and R 2 with L -dR.Then R is a minimal genus surface for the oriented link L if and only if each /^ is a minimal genus surface for the oriented link L z .This generalizes the classical result due to Seifert in the 1930's that the connected sum of minimal genus surfaces is a surface of minimal genus.Corollary 6.9.Let M be a compact connected irreducible oriented 3-manifold whose boundary ΘM is a (possibly empty) union of incompressible tori, and H 2 (M,dM) is not generated by tori and annuli.Then there exists a C 00 transversely oriented foliation ^ on M such that ^ ίίl ΘM, < ΘM has no Reeb components, and no leaf of ^ is compact.In particular we have Corollary 6.11.Let M be either a compact connected oriented 3-manifold whose interior has a complete hyperbolic metric and H 2 (M, ΘM) Φ 0, or M -S 3 -N(L) where L is a nonsplit nontrivial link in S 3 .Then there exists a C°° transversely oriented foliation ¥ of M such that 3F has no compact leaves, ®ί ffl ΘM, and < ΘM has no Reeb components.The conditions that ΘM be a union of incompressible tori and M be irreducible are necessary by Novikov's work.The question of whether a manifold possesses a C°° codimension-1 foliation without compact leaves has been precisely answered by the work of Thurston [31], Levitt [18], Wood [34], and Milnor [19] for circle bundles over surfaces and for most Seifert fibred spaces by [4]; see also [5].The 2-dimensional homology of these spaces (except for trivial cases) is generated by tori and annuli.It would be interesting to