User paul kirk - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T19:21:30Z http://mathoverflow.net/feeds/user/2589 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8924/diffeomorphism-of-3-manifolds/8951#8951 Answer by Paul Kirk for Diffeomorphism of 3-manifolds Paul Kirk 2009-12-15T04:23:41Z 2009-12-15T04:23:41Z <p>Waldhausen proved that homotopy equivalence is homotopic to homeo (and hence diffeo) for Haken 3-manifolds. Perelman extends that to irreducible/infinite pi_1.</p> <p>It's an old conjecture that the Whitehead group of any torsion free group is trivial. </p> <p>Irreducible 3-manifolds either have finite or torsion free pi_1, so given Perelman again only S^3/G have potentially non-simple homotopy equivalences. </p> http://mathoverflow.net/questions/8890/flat-su2-bundles-over-hyperbolic-3-manifolds/8950#8950 Answer by Paul Kirk for Flat SU(2) bundles over hyperbolic 3-manifolds Paul Kirk 2009-12-15T04:05:13Z 2009-12-15T04:05:13Z <p>There is a huge literature on this. I second starting with Klassen's article. You should also go back to Riley's old article and then look at Burde for 2-bridge knots. But I suspect that noone knows an "explicit" description in the sense that "this loop goes to this matrix" in general the representations are given by the real points of a variety defined over Z. </p> http://mathoverflow.net/questions/8924/diffeomorphism-of-3-manifolds/8951#8951 Comment by Paul Kirk Paul Kirk 2009-12-15T23:46:51Z 2009-12-15T23:46:51Z yes, if I'm recalling correctly: if there exist incompressible tori (or surfaces) then Waldhausen applies. Perelman says that atoroidal manifolds are either hyperbolic, so Mostow applies, or (simple) Seifert-fibered (and hence classified), I assume it is known precisely what non-Haken SF manifolds admit h.e. that aren't homotopic to diffeos, presumably just lens spaces.