# Foliation values of a manifold

Let $M$ be a smooth n dimensional manifold. The foliation values of $M$, denoted by $F(M)$, is defined as

$$F(M)=\{ 1\leq k\leq n\mid \text{there exist an smooth k dimensional foliation of M} \}$$

Question:

For $n\in \mathbb{N}$, What is $F(S^{n})$?

• for $n$ even, it is $\{n\}$, because the Euler class implies that the tangent bundle is irreducible. For $n$ odd it always includes $1$. Not sure in about other $k$. – Will Sawin Jan 18 '14 at 23:11
• Reeb constructed (in hs thesis) a codimension 1 foliation of the 3-sphere and from that H. B. Lawson, Jr, constructed codimension 1 foliations on spheres of dimensions $2^k+3$ in [Annals of Mathematics, Second Series, Vol. 94, No. 3 (Nov., 1971), pp. 494-503] So sometimes we do have $n-1$ in $F(S^n)$. Durfee generalized this to odd dimensional spheres. Thurston proved a couple of years later that in fact codimension 1 foliations exists in every closed manifold of Euler characteristic zero. – Mariano Suárez-Álvarez Jan 19 '14 at 2:37
• Ah! Thurston himself tells of all this in a short paper in the Proc. of the ICM 74 in Vancouver. MathSciNet chasing is fun :-) See mathunion.org/ICM/ICM1974.1/Main/icm1974.1.0547.0550.ocr.pdf – Mariano Suárez-Álvarez Jan 19 '14 at 2:48
• $S^{4n-1}$ gets a 3-dimensional foliation from the diagonal free action by $SU(2)$ (or unit quaternions). – Ian Agol Jan 19 '14 at 19:28
• Also, $S^{15}$ is foliated by $S^7$'s, from the fibration $S^7\to S^{15}\to S^8$. – Ian Agol Jan 19 '14 at 20:05