# 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).

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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 4 '11 at 3:58
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 4 '11 at 4:17
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 4 '11 at 5:30
@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 4 '11 at 8:01
@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 4 '11 at 19:57
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## 1 Answer

Borel's conjecture predicts that anu homotopy equivalence of closed aspherical manifolds is homotopic to a homeomorphism. The conjecture has been proved for many fundamental groups, see e.g. this paper 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 this paper by Kruskal-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].

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 this survey 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.

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