Homotopy is not isotopy for homeomorphisms of 3-manifolds (Q1078852)

It has been conjectured for a long time that there is no homeomorphism of a closed 3-manifold which is homotopic but not isotopic to the identity. The authors give in this paper the first counterexample known to this conjecture. This counterexample occurs for the connected sum of two 3- manifolds obtained as quotients of \(S^ 3\) by certain finite subgroups of SO(4); the homeomorphism considered is a Dehn twist along the sphere of connected sum. Motivated by certain problems in quantum gravity, the authors also determine the group of isotopy classes of homeomorphisms fixing a point (resp. a ball) for 3-manifolds of this type.