# Reebless and taut foliations

Suppose we are given a closed oriented 3-manifold. It is well known that taut foliations are Reebless, and if a Reebless foliation isn't taut then the leaves which don't admit a closed transversal are tori. Furthermore, it is straightforward that in a taut foliation all the closed leaves are homologically non trivial.

Conversely, is there any complete criterion to determine if a Reebless foliation is taut?

-

## 1 Answer

For simplicity assume the foliation is transversely oriented (otherwise you can think in terms of the transversely oriented double cover). Therefore each leaf has a canonical orientation. Then a foliation (Reebless or not) is taut if and only if there is no positive linear combination of the oriented torus leaves that is trivial in homology. This is equivalent to the condition that every leaf admits a closed transversal. This basic theory was essentially developed by Novikov in his famous article on the topology of foliations (although Novikov did not develop all the properties of taut foliations).

-
Permit me to supplement with the citation: S. Novikov. "The topology of foliations," Trans. Mosc. Math. Soc., 1965, v. 14, p. 248–278. MR 34:824. mi.ras.ru/~snovikov/23.pdf – Joseph O'Rourke Jun 8 '11 at 12:57