To follow up on [A four-dimensional counterexample?](https://mathoverflow.net/questions/69344/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](http://arxiv.org/pdf/math/0212142)


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).



More four-dimensional counterexamples
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](https://arxiv.org/abs/0901.0442) 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](https://arxiv.org/abs/math/0001063) 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](https://eudml.org/doc/143555)].


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](https://arxiv.org/abs/0902.2480) 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.


