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In the following question, we defined the foliation values of an smooth manifold;

Foliation values of a manifold

Let $S_{i}$'s, $i\in \{0,1,\ldots,27\}$, be the smooth structures of topological $S^{7}$.

According to the above definition, we find foiation values $F_{i}$, where each $F_{i}$ is the foliation values of an smooth manifold, homemorphic to $S^{7}$ with smooth structure $S_{i}$.

The question:

Is $F_{i}=F_{j}$, for all $i,j$? In the other word, is the "Foliation values" a topological invariant?

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  • $\begingroup$ Given that is seems we do not know the foliation values of actual spheres, maybe it is a bit soon to consider the exotic ones? $\endgroup$ – Mariano Suárez-Álvarez Jan 19 '14 at 2:31
  • $\begingroup$ @Mariano thank you for your comment and your valuable information on my related question. Yes, may be it is soon, but note that in this question we do not search for the exact values of $F_{i}$. In fact we are interested to know that whether the smooth structure play an important role?That is, as I said in the question, is "Foliation values" a topological invariant. So I think the nature of this question is different from the nature of the question "What is the EXPLICIT foliation values of the 7 sphere" $\endgroup$ – Ali Taghavi Jan 19 '14 at 10:16
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    $\begingroup$ Some of the exotic spheres are $S^3$-bundles over $S^4$. In this case, clearly $3\in F_i$. In general, I don't know. $\endgroup$ – Sebastian Goette Feb 4 '16 at 13:07
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The answer to your question is yes: For every $k\in\{0,\dots,7\}$, every smooth manifold homeomorphic to $S^7$ admits a $k$-dimensional $C^\infty$ foliation.

Every exotic $7$-sphere $S_i$ is parallelisable. Hence for every $m\in\{0,\dots,7\}$, every $S_i$ admits a $C^\infty$ $m$-frame field (i.e., $TS_i$ admits a trivial $C^\infty$ sub vector bundle of rank $m$). In the case $m>1$, Corollary 1 in W. Thurston: The theory of foliations of codimension greater than one therefore implies that $S_i$ admits a $(7-m)$-dimensional $C^\infty$ foliation. In the case $m=1$, Theorem 1a (or 1b) in W. Thurston: Existence of codimension-one foliations implies that $S_i$ admits a $6$-dimensional $C^\infty$ foliation. The case $m=0$ is trivial.

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  • $\begingroup$ Thank you for your answer. I think In the Paper of Thurston there is a restriction $k<n/2$. yes? $\endgroup$ – Ali Taghavi Aug 13 '17 at 9:13
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    $\begingroup$ @Ali Taghavi: Yes, but that restriction on the codimension $k$ occurs in Corollary 2 (a corollary to Corollary 1), not in the Corollary 1 I cited. Corollary 2 says that in the case $k<n/2$, $S^n$ admits a foliation of codimension $k$ if and only if it admits a $k$-plane field. By e.g. Steenrod: The theory of fibre bundles, Theorem 27.16, $S^n$ admits in the case $k<n/2$ a $k$-plane field (if and) only if it admits a $k$-frame field. Then the existence of a foliation of codimension $k$ follows from Corollary 1. But for $S^7$, we can apply Corollary 1 directly, even without $k<n/2$. $\endgroup$ – Marc Nardmann Aug 13 '17 at 10:52

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