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

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

Question:

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

Ali Taghavi
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  • Your 2nd question doesn't make any sense to me. What does $S^7$ mean to you, as a smooth manifold? – Ryan Budney Jan 18 '14 at 21:40
  • I separate the second question(I delete it from this question and give it in a new question, with more explanation) – Ali Taghavi Jan 18 '14 at 22:40
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    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
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    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 02:37
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    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 http://www.mathunion.org/ICM/ICM1974.1/Main/icm1974.1.0547.0550.ocr.pdf – Mariano Suárez-Álvarez Jan 19 '14 at 02:48
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    $S^{4n-1}$ gets a 3-dimensional foliation from the diagonal free action by $SU(2)$ (or unit quaternions). – Ian Agol Jan 19 '14 at 19:28
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    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
  • @IanAgol What about $\mathbb{C}P^2$? Does it admit a 2 dimensional foliation? – Ali Taghavi Oct 17 '17 at 14:19
  • @IanAgol My apology for this second comment. if the answer to the previous comment is yes, does it admit a foliation whose leaves are totally geodesics with respect to Fubbini Study metric? – Ali Taghavi Oct 17 '17 at 17:07

0 Answers0