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


[Foliation values of a manifold](https://mathoverflow.net/questions/154975/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?
> 
> 
> 



Foliation values of Exotic spheres
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.](https://mathoverflow.net/questions/58131) 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*](https://eudml.org/doc/139581) 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*](https://www.jstor.org/stable/1971047) implies that $S\_i$ admits a $6$-dimensional $C^\infty$ foliation. The case $m=0$ is trivial.


