Are there examples of smooth closed manifolds (not necessarily oriented) that admit an open book decomposition but that are not the boundary of any compact smooth manifold?



open book decompositions and being a boundary
In dimensions > 6 a simply connected manifold is an open book if and only if its signature is zero. (winkelnkemper '73)


On the other hand, for example the 8-dimensional bordism group is $Z\oplus Z$ (generated by the simply connected manifolds $CP^4$ and $CP^2\times CP^2$), so it necessarily has some nontrivial (and simply connected) element of signature 0.


