Classification of trisections and the Generalized Property R Conjecture (Q2821758)

We call \(\natural^G(S^1\times D^{n-1})\) a \textit{genus \(G\), \(n\)-dimensional 1-handlebody}. \textit{Trisection} is a decomposition of a closed orientable smooth 4-manifold \(X\) into \(X=X_1\cup X_2\cup X_3\), where \(X_i\) is a genus \(k_i\) 4-dimensional 1-handlebody and satisfies the following conditions:NEWLINENEWLINE(1) If \(i\neq j\), then \(X_i\cap X_j\) is homeomorphic to a genus \(g\) 3-dimensional 1-handlebody.NEWLINENEWLINE(2) \(X_1\cap X_2\cap X_3\) is a genus \(g\) closed surface \(\Sigma_g\).NEWLINENEWLINEThe genus \(g\) is called \textit{genus of the trisection}. This decomposition is called a \textit{\((g;k_1,k_2,k_3)\)-trisection}. If \(k_1=k_2=k_3\), then the trisection is called \textit{balanced}, otherwise, \textit{unbalanced}.NEWLINENEWLINEIn [Geom. Topol. 20, No. 6, 3097--3132 (2016; Zbl 1372.57033)], \textit{D. T. Gay} and \textit{R. Kirby} proved that every closed orientable smooth 4-manifold admits a balanced trisection. This result is the starting point of trisection theory of 4-manifolds. Trisections were relatively recently (early in the 2010s) defined and developed, and it has been expected that through some kind of relationship between 4-manifold theory and Heegaard decomposition theory, many problems in both theories may be brought closer to solution. One candidate for this may be the Generalized Property R Conjecture:NEWLINENEWLINE[The Generalized Property R Conjecture] Suppose that \(L\) is a \(c\)-component link in \(S^3\) with an integral surgery to \(\#^c(S^1\times S^2)\). Then there is a sequence of handleslides transforming \(L\) into a \(c\)-component unlink.NEWLINENEWLINE\(S^4\) is the only 4-manifold admitting a genus zero trisection. There are three balanced genus one trisections, corresponding to \(\mathbb {C}P^2, \overline{\mathbb {C}P^2}\) and \(S^1\times S^3\). In addition, there are three unbalanced trisections of genus one corresponding to the three stabilizations of the genus zero trisection of \(S^4\). This article gives the following classification.NEWLINENEWLINE(Main result) Let \(X\) be a 4-manifold admitting a \((g;k_1,k_2,k_3)\)-trisection \(\mathcal{T}\). Let \(k'\) be \(\min\{k_2,k_3\}\). If \(k_1\geq g-1\), then \(X\) is diffeomorphic either to \(\#^{k'}(S^1\times S^3)\) or to the connected sum of \(\#^{k'}(S^1\times S^3)\) with one of \(\mathbb {C}P^2\) or \(\overline{\mathbb {C}P^2}\), and \(\mathcal{T}\) is the connected sum of genus one trisections.NEWLINENEWLINEAn important ingredient to prove this main result are results on the classification of Dehn surgeries between connected sums of \(S^1\times S^2\). The main theorem, in turn, again yields Dehn surgery results that are applicable to the Generalized Property R Conjecture:NEWLINENEWLINE(Corollary 1.3.) Suppose that \(L\) is a \(c\)-component link in \(\#^k(S^1\times S^2)\) with an integral framed surgery to \(\#^{c+k}(S^1\times S^2)\).NEWLINENEWLINE(1) If \(L\) has tunnel number \(c+k-1\), then \(L\) is a \(c\)-component \(0\)-framed unlink.NEWLINENEWLINE(2) If \(L\) has tunnel number \(c+k\), then there is a sequence of handleslides taking the split union of \(L\) with a \((k+1)\)-component \(0\)-framed unlink to a \((c+k+1)\)-component 0-framed unlink.NEWLINENEWLINEIn the case of \(k=0\), this result implies that any \(c\)-component link with tunnel number \(c-1\) satisfies the Generalized Property R Conjecture.