Tight foliage

In mathematics , especially in differential geometry and topology are taut foliations (ger .: taut foliations ) foliations that are characterized by minimal surfaces a suitable Riemannian metric can be realized.

definition

Be a manifold . A scroll of codimension 1 is called tight if there is an image for each leaf , the image of which intersects transversely . ${\displaystyle M}$ ${\displaystyle L}$ ${\displaystyle f\colon S^{1}\rightarrow M}$ ${\displaystyle L}$

Realizability through minimal areas

Be a closed, oriented, differentiable manifold. According to a theorem of Rummler and Sullivan, the following conditions are equivalent to a transversely orientable codimension 1 foliation : ${\displaystyle M}$ ${\displaystyle {\mathcal {F}}}$

• ${\displaystyle {\mathcal {F}}}$ is tight
• there is a too transverse flow that leaves a volume shape invariant ${\displaystyle {\mathcal {F}}}$
• there is a Riemannian metric in which the leaves of surfaces are the smallest area. ${\displaystyle M}$ ${\displaystyle {\mathcal {F}}}$

Leaves without Reeb components

If a foliage is tight, there cannot be a Reeb component , i.e. H. do not give a subset diffeomorphic to a Reeb foliage . The reverse also applies to atoroidal 3-manifolds: every foliage without Reeb components is tight.

Tight foliage of 3-manifolds

There is a well-developed structure theory for tight scrolling of 3-manifolds. First of all, according to Novikov-Zieschang's theorem, tight foliage can only exist on a closed, orientable 3-manifold if or , and then all the leaves must necessarily be incompressible . Gabai's theorem provides a sufficient condition for the existence of tight foliage : Let M be a closed, irreducible 3-manifold with , then there is a tight foliage on M. One can even realize every nontrivial element of as a leaf of a tight foliage. Gabai's proof uses grained manifold hierarchies . ${\displaystyle \pi _{2}(M)=0}$ ${\displaystyle M=S^{2}\times S^{1}}$ ${\displaystyle H_{2}(M)\not =0}$ ${\displaystyle H_{2}(M)}$

Palmeira's theorem provides an approach to the structure of tight foliage on 3-manifolds: If there is a tight foliage on a closed, orientable 3-manifold , then the universal overlay is diffeomorphic to and the raised foliage is a foliation of the foliage diffeomorphic to . The space of the leaves (of the raised foliage) is in this case a (generally non-Hausdorffian) 1-manifold and the tight foliage is thus described by an action of on a 1-manifold. ${\displaystyle M\not =S^{2}\times S^{1}}$ ${\displaystyle {\widetilde {M}}}$ ${\displaystyle R^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \pi _{1}M}$

L-spaces do not have tight leaves.

Web links

• Manifold Atlas
• Kazez-Roberts: Taut foliations

supporting documents

1. ↑ Sullivan, Dennis A homological characterization of foliations consisting of minimal surfaces. Comment. Math. Helv. 54 (1979), no. 2, 218-223, doi: 10.1007 / BF02566269 .
2. ↑ Novikov, SP: Топология слоений. Тр. Моск. мат. о-ва. 14 1965. 248-278.
3. ↑ Gabai, David: Foliations and the topology of 3-manifolds. J. Differential Geom. 18 (1983) no. 3, 445-503. online (pdf)
4. ^ Palmeira, Carlos Frederico Borges: Open manifolds foliated by planes. Ann. Math. (2) 107 (1978) no. 1, 109-131. online (pdf)