Open book decompositions in dimension 4

Question

The question about the existence of open book decompositions for a closed oriented $n$-dimensional manifold seems to be answered in all dimensions except dimension $4$, where as far as I can tell the situation seems to be quite mysterious. By playing around with the definition one can see that the signature has to vanish and the Euler characteristic has to agree with the Euler characteristic of a surface. Since the situation does not seemed to be resolved yet, these are probably not the only restrictions.

So my questions are:

1. What is the state of the art about open book decompositions in dimension $4$?

2. Do there exist requirements that ensure that a $4$-manifold admits an open book decomposition

3. Do there exist requirements that ensure that a $4$-manifold admits an open book decomposition up to stabilization i.e. connected sum with certain other $4$-manifolds

4. As a non-trivial example, does the product of two surfaces of genus at least $2$ admit an open book decomposition? Does it admit one up to stabilization?

Answer 1

Nice question; there's not much known in general. The signature obstruction you mention goes back to Winkelnkemper (Bull. Amer. Math. Soc. 79 (1973), 45–51) and is sufficient for simply connected $4n$-manifolds when $n>1$. The general high-dimensional case there are further obstructions; see Quinn (Topology 18 (1979), no. 1, 55–73), Neumann (Topology 14 (1975), no. 3, 237–244) and Lawson (Topology 17 (1978), no. 2, 189–192). It's possible that Quinn's obstructions are stable obstructions; you'd have to read the paper to find out.

For your question 3: Simply connected $4$-manifolds of signature $0$ are stably diffeomorphic to connected sums of $S^2 \times S^2$ or the twisted bundle $S^2 \tilde{\times} S^2$; those connected sums have open book decomposition. So simple connectivity would suffice.

Finally, there's an interesting connection between open books and Engel structures in dimension $4$; see Colin-Presas-Vogel (Algebraic & Geometric Topology 18 (2018) 4275–4303).

Answer 2

Adding to the answer of Danny Ruberman: One source that summarizes the state of the art pretty well is given by Andrew Ranicki in High dimensional knot theory. In Chapter 29 he reformulates the obstruction by Quinn in terms of surgery theory. His invariant, called asymmetric signature $\sigma^*$, refines the signature invariant for non-simply connected manifolds. It is proven in all dimensions including dimension 4 that if an open book exists then $\sigma^*$ vanishes. Also in all dimensions except dimension 4, that the reverse implication holds. Using that formulation it easier to see that $\sigma^*$ is in fact a stable invariant up to stabilization with $S^2 \times S^2$. With Quinn's original more geometric approach it was not as easy to see that.

We know that Quinn's proof to show the reverse implication does not work in dimension 4 for several reasons. However the statement might still be true. We have a proof outline with Marc Kegel which we are working out at the moment to show that any 4 manifold with vanishing asymmetric signature $\sigma^*$ carries an open book after stabilization.