Doubles without open book decompositions from higher signatures

TL;DR

This paper demonstrates the existence of closed manifolds that are doubles but lack open book decompositions in all even dimensions, revealing new insights into high-dimensional topology and the signature's behavior.

Contribution

It provides the first examples of doubles without open book decompositions in all even dimensions, challenging previous assumptions in high-dimensional knot theory.

Findings

Existence of doubles without open book decompositions in all even dimensions

Counterexamples to high-dimensional knot theory conclusions

Connection between signature non-multiplicativity and manifold properties

Abstract

We show that in every even dimension there are closed manifolds that are doubles, but have no open book decomposition. In high dimensions, this contradicts the conclusions in Ranicki's book on high-dimensional knot theory. In all dimensions, examples arise from the non-multiplicativity of the signature in fibre bundles. We discuss many examples and applications in dimension four, where this phenomenon is related to the simplicial volume.