Type: Seminar

Distribution: World

Expiry: 23 Mar 2007

Auth: jonh@asti.maths.usyd.edu.au

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.