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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).

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Igor Rivin
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    Borel's conjecture predicts that a homotopy equivalence of closed aspherical manifolds is homotopic to a homeomorphism. I would be very surprized if the conjecture were not true for surface bundles over surfaces. – Igor Belegradek Jul 04 '11 at 03:58
  • Incidentally, if both the fiber and the base have zero Euler characteristic, then I think the fundamental group of the total space has subexponetial growth in which this case the Borel's conjecture is true. – Igor Belegradek Jul 04 '11 at 04:17
  • Cool! I would assume that "Borel's conjecture" would be based on some set of cases where Borel actually knew this to be true... Also, what's the reference for the "subexponential growth" case? – Igor Rivin Jul 04 '11 at 05:30
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    @Igor: "if both the fiber and the base have zero Euler characteristic, then I think the fundamental group" --- what about $T^2$-bundle over $S^1$? – Anton Petrunin Jul 04 '11 at 08:01
  • @Anton: thanks, I wrote this in haste. In fact, I do not even know the 4-dimensional Borel conjecture for all subexponential growth groups, but all virtually nilpotent groups are okay. I will give references later today. – Igor Belegradek Jul 04 '11 at 19:57
  • I believe the canonical reference is Freedman-Quinn (for polycyclic groups) (in "Topology of 4-manifolds") – Igor Rivin Jul 04 '11 at 20:49

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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.

Kevin Walker
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