Joint Colloquium: Hillman -- Finiteness conditions and mapping tori

The mapping torus of a self-homeomorphism $f$ of a space $X$ is the space
obtained from the cylinder $Xx[0,1]$ by identifying the ends via $f$. (The
Moebius band is a simple, nontrivial example of this construction.) There are
good characterizations of $n$-manifolds which are mapping tori in all
dimensions except $n=4$ or 5. We shall consider a homotopy analogue of the
problem of recognizing mapping tori, in which "homeomorphism" and "manifold"
are replaced by "homotopy equivalence" and "Poincare duality complex". Our
argument is homological, and an essential element is Kochloukova’s recent 
result that certain Novikov extensions of group rings are weakly finite.

Our results are best possible, in a sense to be explaned in the talk. In
particular, we obtain the following 4-dimensional analogue of Stallings’
characterization of 3-dimensional mapping tori:

If $M’$ is an infinite cyclic covering space of a closed 4-manifold $M$ then
$M’$ satisfies 3-dimensional Poincare duality with local coefficients if and
only if $\chi(M)=0$ and $\pi_1(M’)$ is finitely generated.