3-fold of general type homeomorphic to rational 3-fold Is there a smooth (complex projective) 3-fold of general type which is homeomorphic (in the complex topology) to a rational $3$-fold? 
I am aware of such examples in complex dimension $2$, for example the Barlow surface is homeomorphic to a blow up $\mathbb{P}^{2}$ in 8 points.
 A: This is not a complete solution. We prove that 69 out of the 105 smooth Fano $3$-folds are not homeomorphic to a 3-fold of general type.
The proof is similar to Sai-Kee Yeung's proof of the non-existence of a "fake $\mathbb{CP}^{3}$" (Theorem 3.1 of http://journals.math.ac.vn/acta/pdf/1001199.pdf), which I learned about is dhy's  comment here 3-folds with "simple" Betti numbers and positive Kodaira dimension and relies on the Miyaoka-Yau inequality.
In particular, we prove the following: There is no smooth $3$-fold of general type with $b_{1}=0$ and $b_{3} \leq 2$.
Proof: 
By the assumption $b_{1}=0$, $h^{1,0}=0$. Then by applying the Hirzbruch Riemann-Roch theorem to $\mathcal{O}_{X}$ we obtain:$$\frac{c_{1}c_{2}(X)}{24} = \chi(\mathcal{O}_{X}) = 1 + h^{2,0} - h^{3,0}$$. Since $b_{3} \leq 2$, $h^{3,0} \leq 1$ and hence $c_{1}c_{2}(X) \geq 0$. Then by the Miyaoka-Yau inequality for smooth $3$-folds $(c_{1})^{3} \geq \frac{8 c_{1}c_{2}}{3} \geq 0$ contradicting the fact that $K_{X}$ is ample. (The exact form of the Yau inequality we use is that same as in Yeung's paper, namely that $8c_{2}(X)- 3c_{1}^{2}(X)$ is a pseudo-effective class).QED. 
To obtain the statement about Fano's, it is well-known that smooth Fano's satisfy $b_{1}=0$ (they are even simply connected), and looking at the Ivskovskikh-Mori-Mukai list, we see that 69 families satisfy $b_{3} \leq 2$, in particular no smooth Fano $3$-fold with Picard rank at least $4$ is homeomorphic to a smooth $3$-fold of general type.  By Miyaoka's generalisation of Yau's inequality the class $3c_{2}-c_{1}^{2}$ is pseudo-effective for $3$-folds with $K_{X}$ nef and big, so the above applies to this class also.
