$L^2$-invariants and fibering over $S^1$: aspherical counterexamples and new secondary obstructionsMPIM

by Wolfgang Lück (Universität Bonn)

Thursday Mar 26, 2026, 1:30 PM → 3:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Oberseminar Differentialgeometrie

Kielak showed that for RFRS-groups the vanishing of the first $L^2$-Betti number is the only obstruction to algebraic fibering, motivated by the proof of Agol of the Virtually Fibering Theorem.
Fisher extended this to higher dimensions for the condition FP(Q).  This raises the questions whether the vanishing of $L^2$-Betti numbers may be sufficient for fibering in some special cases, e.g., closed aspherical manifolds or hyperbolic manifolds.

We will show that there exist examples of connected closed aspherical smooth manifolds with residual (torsion-free nilpotent) fundamental group for which all $L^2$-Betti numbers vanish and which do not virtually fiber over $S^1$.

We also present  new secondary obstructions to fibering based on twisted $L^2$-torsion and universal $L^2$-torsion.

This is  an ongoing project joint with Sam Hughes and Ian Leary.