Fibration of the $n$-holed torus Let $X$ be the $n$-holed torus, a $2$-dimensional manifold.
For $n = 0$, there is a fibration $S^3 \rightarrow X \cong S^2$ with fibers $S^1$.
For $n = 1$, there is a fibration $\mathbb{R}^2 \rightarrow X \cong \mathbb{T}^2$ to the $2$-dimensional torus. Its fibers are discrete (dimension $0$) and in correspondence with $\mathbb{Z}^2$.
For other $n > 1$, I am looking for manifolds $M_n$ with $\pi_0(M) = 0, \pi_1(M) = 0, \pi_2(M) = 0$, and a fibration $M \rightarrow X$, whose fibers are $1$-dimensional manifolds (homeomorphic to either $\mathbb{R}$ or $S^1$ or disjoint unions of these).
If there aren't any, then are there higher dimensional manifolds with such fibrations?
 A: Let me use the more standard notation $\Sigma_g$ for the closed orientable surface of genus $g$. Applying the long exact sequence in homotopy to a fibration $F \to M \to \Sigma_g$ with $M$ satisfying your conditions gives first an isomorphism
$$\pi_1(\Sigma_g) \cong \pi_0(F)$$
and second an isomorphism
$$\pi_2(\Sigma_g) \cong \pi_1(F).$$
Since the universal cover of $\Sigma_g$ is (the hyperbolic plane $\mathbb{H}$ therefore) contractible $\pi_2(\Sigma_g)$ vanishes. We conclude that if $F$ is a $1$-manifold then it must be a disjoint union of $|\pi_1(\Sigma_g)|$ copies of $\mathbb{R}$. We can get such a fibration by taking $M = \mathbb{H} \times \mathbb{R}$ where the map $M \to \Sigma_g$ is the composite of the projection $\mathbb{H} \times \mathbb{R} \to \mathbb{H}$ with the universal covering map $\mathbb{H} \to \Sigma_g$.
This is sort of "boring," though, since we just slotted in the $\mathbb{R}$ by fiat. Arguably a more natural construction is the following. If we equip $\Sigma_g$ with a hyperbolic metric of constant curvature then we can consider its unit tangent bundle $UT(\Sigma_g)$, a space which naturally sits in a nontrivial fibration
$$S^1 \to UT(\Sigma_g) \to \Sigma_g.$$
(When applied to the sphere this construction produces the Hopf fibration.) $UT(\Sigma_g)$ is naturally the quotient of the unit tangent bundle $UT(\mathbb{H}) \cong PSL_2(\mathbb{R})$ of the hyperbolic plane by the action of $\pi_1(\Sigma_g)$ acting by hyperbolic isometries, and then one can further contemplate taking the universal cover of this space, which is contractible. Altogether this should give a fibration
$$F \to \widetilde{PSL}_2(\mathbb{R}) \to \Sigma_g$$
with the desired properties which is somewhat more "natural," although it may actually be homotopy equivalent as a fibration to the previous one, I don't know.
