More four-dimensional counterexamples - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T20:21:49Z http://mathoverflow.net/feeds/question/69431 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69431/more-four-dimensional-counterexamples More four-dimensional counterexamples Igor Rivin 2011-07-04T00:36:29Z 2011-07-04T21:15:43Z <p>To follow up on <a href="http://mathoverflow.net/questions/69344/a-four-dimensional-counterexample" rel="nofollow">http://mathoverflow.net/questions/69344/a-four-dimensional-counterexample</a>, 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 <a href="http://arxiv.org/pdf/math/0212142" rel="nofollow">Hillman's book</a></p> <p>he seems very careful to sidestep this question and talk about homotopy equivalence only...</p> <p>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).</p> http://mathoverflow.net/questions/69431/more-four-dimensional-counterexamples/69498#69498 Answer by Igor Belegradek for More four-dimensional counterexamples Igor Belegradek 2011-07-04T21:15:43Z 2011-07-04T21:15:43Z <p>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. <a href="http://front.math.ucdavis.edu/0901.0442" rel="nofollow">this paper</a> by Bartels-Lueck.</p> <p>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 <a href="http://front.math.ucdavis.edu/0001.5063" rel="nofollow">this paper</a> by Kruskal-Quinn.</p> <p>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]. </p> <p>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.</p> <p>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 <a href="http://front.math.ucdavis.edu/0902.2480" rel="nofollow">this survey</a> 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.</p>