Doubles without open book decompositions from higher signatures

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.

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