Structure sets vanish for certain bundles over Seifert manifolds

Abstract:

Let ${M^{n + 3}}$ be a compact orientable manifold which is the total space of a fiber bundle over a compact orientable manifold ${K^3}$ with an effective circle action of hyperbolic type. Assume that the fiber ${N^n}$ 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 ${S_{\text {TOP}}} ({N^n} \times {D^k},\partial )$ of topological surgery vanishes for sufficiently large $k$. Then the projective class and Whitehead groups of $M$ vanish and ${S_{\text {TOP}}} ({M^{n + 3}} \times {D^k},\partial ) = 0$ if $n + k \geqslant 3$ or if ${K^3}$ is closed and $n = 2$. The $\text {UNil}$ groups of Cappell are the main obstacle here, and these results give new examples of generalized free products of groups such that ${\text {UNil}}_j$ vanishes in spite of the failure of Cappell’s sufficient condition.