Homotopy is not isotopy for homeomorphisms of 3-manifolds

Abstract

existence of a homeomorphism that is homotopic but not isotopic to the identity has remained an open question for closed 3-manifolds [I, 23. We consider here homeotopy groups:: of spherical spaces, finding as a by-product of our work an example of such a homeomorphism for a closed 3-manifold whose prime factors include certain spherical spaces. The homeotopy groups of a composite 3-manifold have as subgroups the disk-fixing or point-fixing homeotopy groups of each prime factor [3,4]. In the work reported here our primary aim has been to calculate, for spherical spaces the corresponding 0th homeotopy groups, the groups of path connected components of the spaces of disk-fixing and point- fixing homeomorphisms. Homeomorphism groups of spherical spaces have been considered recently by Rubinstein et al. [S-7]. Asano [8], Bonahon [9] and Ivanov [lo]. Their results are consistent with Hatcher’s conjecture [ 1 l] that for each spherical space the group of homeomorphisms has the same homotopy type as the group of isometries. Homotopy classes of the groups HO and XX of homeomorphisms that fix respectively a disk and a point do not generally have this character (for spherical spaces): in particular, nonzero elements of ~,,(&‘a) and rr,, (XX) are commonly not represented by isometries. For several spherical spaces of the form