Doubles without open book decompositions from higher signatures
D. Kotschick
TL;DR
The paper shows that in every even dimension there exist doubles that admit no open book decomposition, contradicting high-dimensional claims. The obstruction comes from the non-multiplicativity of the signature in fibre bundles, detected by the Quinn invariant, with explicit constructions built from surface bundles over surfaces and products of high-genus surfaces. In dimension four, the phenomenon is tied to the simplicial volume: open books force zero simplicial volume, while many examples with non-multiplicative signatures have positive simplicial volume, including hyperbolic ball quotients and their blowups. The results yield many Engel four-manifolds without open books, providing a negative answer to whether all Engel structures are supported by open books, and illuminate connections to SK-groups and asymmetric signatures, correcting previous claims in Ranicki’s work.
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.