More four-dimensional counterexamples To follow up on A four-dimensional counterexample?, I am probably being dense, but are there examples of spaces which are homotopy equivalent to bundles of surfaces over surfaces (or three-manifolds over the circle, or circle over three-manifold) and yet are not such. Same question if you change "bundle" to "product". In Hillman's book
he seems very careful to sidestep this question and talk about homotopy equivalence only...
I am interested primarily in spaces where the fiber and the base are $K(\pi, 1)$ spaces (so not spheres) and are oriented (if that makes any difference).
 A: Borel's conjecture predicts that any homotopy equivalence of closed aspherical manifolds is homotopic to a homeomorphism. The conjecture has been proved for many fundamental groups, see e.g.
The Borel Conjecture for hyperbolic and CAT(0)-groups by Bartels-Lueck.
Basic ingredients are topological surgery, and computations of $L$-groups and $K$-groups.
Surgery works  perfectly in dimensions $\ge 5$, but in dimension $4$ one is limited to fundamental groups of subexponential growth, see
Subexponential groups in 4-manifold topology by Krushkal-Quinn.
I am not up to date with L-theory computations but for polycyclic groups the reference is
[Farrell-Jones, The surgery L-groups of poly-(finite or cyclic) groups, Invent. Math. 91 (1988), no. 3, 559–586, EuDML].
Combining the two ingredients gives Borel's conjecture for closed 4-manifolds homotopy equivalent to an infranil 4-manifold. This includes direct products of two closed surfaces with zero Euler characteristic.
Looking at recent papers by Bartels, Farrell, and Lueck at arxiv will bring you to state of the art on the L-theory computations, in particular, see
Survey on aspherical manifolds by Lueck,
but I am not aware of the result that covers surface bundles specifically, and in any case one is limited to groups of subexponential growth.
