Trisecting a 4-dimensional book into three chapters (Q6584168)

A trisection \(\mathcal{T}\) of a closed 4-manifold \(W\) is a decomposition \(W=W_0 \cup W_1 \cup W_2\), where each \(W_i\) is a 4-dimensional 1-handlebody \(\natural _k(S^1 \times B^3)\), and for \(i \neq j\), \(W_i \cap W_j\) is a 3-dimensional 1-handlebody \(\natural _g(S^1 \times D^2)\), and \(W_0 \cap W_1 \cap W_2\) is a closed surface of genus \(g\). An open book decomposition of \(W^4\) is a submanifold \(B^2\) of codimension 2 with trivial normal bundle in \(W\) and a trivial fibration \(p: W\backslash B \to S^1\) that looks on a tubular neighborhood \(\nu B\) of \(B\) like the standard angular fibration \(p: \nu B \backslash B \cong B \times (D^2 \backslash \{0\}) \to S^1\). The page of an open book decomposition is the closure of the preimage \(M=\mathrm{Cl}(p^{-1}(\phi))\), that is a compact 3-manifold. An open book decomposition induces an abstract open book, that is obtained from the page \(M\) and the monodromy, where the monodromy is a certain conjugacy class in the mapping class group of \(M\). These notions extend to 4-manifolds with boundary, as a relative trisection and an open book decomposition. \newline Let \(W\) be a compact, smooth and oriented 4-manifold. The main result is that if \(W\) is closed (respectively has non-empty boundary) and admits the structure of an abstract open book, then \(W\) carries a \((g,k)\)-trisection (respectively a relative \((g,k; p,b)\)-trisection) where \(g,k\) (respectively \(g,k,p,b\)) are determined from the structure. The authors give an algorithm to create an explicit trisection diagram from an open book decomposition. Similar results hold true for \(W\) which admits the structure of a fiber bundle over \(S^1\) with fiber \(M\). The authors give examples including surface bundles over \(S^2\), the spuns of the Lens spaces, and the twist spun manifold of the Poincaré homology sphere. For a preliminary for the proof, the authors discuss certain mapping class groups of compression bodies, adapted to Heegaard splittings. The Heegaard splitting of the page induces a splitting of the 4-dimensional open book into two open books. Using the trisection of the part coming from the 1-handlebody of the Heegaard splittings, and the traces of the handle maps, the authors explicitly construct the spine of the trisection.