I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do **not** admit any contact structure. 
Can someone provide me with some examples? 



Examples of odd-dimensional manifolds that do not admit contact structure
According to a well known result of Martinet, every compact orientable $3$-dimensional manifold has a contact structure [2], see also [1] for various proofs. On the other hand we have



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> **Theorem.** For $n\geq 2$ there is a closed oriented connected manifold of dimension $2n+1$ without a contact structure.
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For $n=2$, $SU(3)/SO(3)$ has no contact structure and for $n>2$,
$SU(3)/SO(3)\times\mathbb{S}^{2n-4}$ has no contact structure, see Proposition 2.4 in [3].


**[1] H. Geiges,** *An introduction to contact topology*, Cambridge studies in advanced mathematics 109.


**[2] J. Martinet,**
Formes de contact sur les variétés de dimension 3. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 142–163. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971. 


**[3] R. E. Stong,** Contact manifolds. *J. Differential Geometry* 9 (1974), 219–238. 


