Structure sets vanish for certain bundles over Seifert manifolds

Abstract

Abstract. Let M" + i be a compact orientable manifold which is the total space of a fiber bundle over a compact orientable manifold K3 with an effective circle action of hyperbolic type. Assume that the fiber N" in this bundle is a closed orientable manifold with Noetherian integral group ring, with vanishing projective class and Whitehead groups, and such that the structure set SJOP(N" X Dk, 3) of topological surgery vanishes for sufficiently large k. Then the projective class and Whitehead groups of M vanish and ST0P(M"+3 X Dk, 3) = 0 if n + k> 3 or if AT3 is closed and n = 2. The UNil groups of Cappell are the main obstacle here, and these results give new examples of generalized free products of groups such that UNily vanishes in spite of the failure of Cappell's sufficient condition.A conjecture of long standing holds that a closed aspherical manifold is determined up to homeomorphism by its fundamental group. This claim is now known to be valid for aspherical manifolds of dimensions greater than four whose fundamental groups contain nilpotent subgroups of finite index [8,10]; earlier work verified the conjecture for poly-Z fundamental groups, subject to the same dimension restrictions [30, pp. 228-231]. This paper will show that the conjecture and related statements are valid for compact manifolds M which are total spaces of certain bundles over surfaces or 3-manifolds. The fibers of these bundles are closed, orientable manifolds N satisfying these two hypotheses: