To follow up on http://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][1]


  [1]: 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).