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It is well known that there exist infinitely many (non homeomorphic) non-Haken closed hyperbolic 3-manifolds. These can be obtained for example doing Dehn surgery on the figure eight knot complement. However, these manifolds may contain embedded nonorientable closed surfaces.

So, my question is:

Are there infinitely many (non homeomorphic) non-Haken closed hyperbolic 3-manifolds, which does not contain embedded nonorientable closed surfaces?

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Every non-orientable closed surface $\Sigma \subset M$ in an orientable 3-manifold $M$ determines a non-trivial element $[\Sigma]\in H_2(M,\mathbb Z/_{2\mathbb Z})$. The element is non-trivial because the complement $M\setminus \Sigma$ is connected.

Therefore if $M$ is a homology sphere it contains no non-orientable closed surface. Every $1/n$-surgery on the figure-eight knot is a homology sphere, hence there are infinitely many manifolds that work, so the answer is yes.

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