On 4-dimensional 2-handlebodies and 3-manifolds (Q2920397)

This article resolves two problems in low-dimensional topology. One is the Montesinos-Fox conjecture, which it is a long-standing problem concerning branched coverings. The other is Kerler's conjecture concerning the cobordism category of 3-manifolds. The Montesinos-Fox conjecture is the following:NEWLINENEWLINEProblem A. Two labeled link diagrams represent the same 3-manifold as a simple branched covering over \(S^3\), if and only if the diagrams can be related to each other by a finite sequence of labeled isotopy moves and Montesinos moves.NEWLINENEWLINEThis conjecture was proven in the case of 3-fold coverings up to 4-fold stabilization [\textit{R. Piergallini}, Topology 34, No. 3, 497--508 (1995; Zbl 0869.57002)], and in the case of 4-fold coverings up to 5-fold stabilization [\textit{N. Apostolakis}, Algebr. Geom. Topol. 3, 117--145 (2003; Zbl 1014.57001)]. Unlike these proofs, the present authors prove Problem A in terms of cobordism category.NEWLINENEWLINEKerler's conjecture is the following:NEWLINENEWLINEProblem B. The category \(\widetilde{Cob}^{2+1}\) of 2-framed relative 3-dimensional cobordisms is equivalent to some universal monoidal braided category freely generated by a Hopf algebra object.NEWLINENEWLINEKerler constructed a universal monoidal braided category \({\mathcal Alg}\) freely generated by a Hopf algebra object and a full functor \({\mathcal Alg}\to \widetilde{Cob}^{2+1}\) [\textit{T. Kerler}, Contemp. Math. 318, 141--173 (2003; Zbl 1168.57313)]. He, as a challenging problem, posted the problem to find additional relations on \({\mathcal Alg}\) to induce a category equivalence. In other words, to find an algebraic characterization of the 3-dimensional cobordism category.NEWLINENEWLINETo resolve Problems A and B, the authors show an equivalence functor between two categories NEWLINE\[NEWLINE{\mathcal S}^c_n\to {\mathcal Chb}^{3+1}={\mathcal Chb}^{3+1}_1NEWLINE\]NEWLINE for any \(n\geq 4\), and a factorization NEWLINE\[NEWLINE\Theta_n:{\mathcal S}^c_n\to {\mathcal H}^{r,c}_n\to {\mathcal K}_n^c.NEWLINE\]NEWLINENEWLINENEWLINEThe category \({\mathcal Chb}^{3+1}_n\) is the one of relative 4-dimensional 2-handlebodies between 3-dimensional 1-handlebodies with \(n\) 0-handles modulo 2-equivalence (handle sliding, creating/canceling of 1-/2-handle pair). This paper, first, defines a cobordism category \({\mathcal Chb}^{3+1}_n\) and a braided category \({\mathcal K}_n\) of Kirby tangles, and shows that these categories are equivalent to each other through the presentation of Kirby diagrams of 4-dimensional 2-handlebodies.NEWLINENEWLINEThen, the authors define the category \({\mathcal S}_n\) of \(n\)-labeled ribbon surfaces between regularly embedded arcs in the 3-dimensional Euclidean space, modulo 26 labeled 1-isotopy moves and 2 ribbon movesNEWLINENEWLINEThese categories \({\mathcal Chb}^{3+1}_n\), \({\mathcal K}_n\), and \({\mathcal S}_n\) are strict monoidal categories. The associative product \(\diamond\) is defined to be the process connecting the two \(n\)-labeled handlebodies by \(n\) 1-handles. The authors give a functor \(\Theta_n:{\mathcal S}_n\to {\mathcal K}_n\) by taking the simple branched covering along the ribbon surface and the restriction \(\Theta_n:{\mathcal S}_n^c\to {\mathcal K}_n^c\) gives a category equivalence as a braided monoidal functor, where the superscript \(c\) presents connected handlebodies.NEWLINENEWLINEAn algebraic category \({\mathcal H}^r_n\) is a strict monoidal braided category for the groupoid \({\mathcal G}_n=\{1,\cdots,n\}^2\). The authors define two functors \(\Phi_n:{\mathcal H}_n^{r,c}\to {\mathcal K}_n^c\) and \(\Psi_n:{\mathcal S}_n^c\to {\mathcal H}^{r,c}_n\) such that \(\Theta_n=\Phi_n\circ \Psi_n\) and show \(\Phi_n,\Psi_n\) are equivalent. \(\Psi_n\) is an algebraic interpretation of simple branched covering representation of 4-dimensional relative 2-handlebody cobordism.NEWLINENEWLINEThe category \(\widetilde{Cob}^{2+1}_n\) is defined as the quotient of \({\mathcal Chb}^{3+1}_n\) under 1/2-handle trading, and \(Cob^{2+1}_n\) is the quotient of \({\mathcal Chb}_n^{3+1}\) under 1/2-handle trading and blowing down/up. These give the quotient functors: NEWLINE\[NEWLINE{\mathcal Chb}^{3+1}_n\to \widetilde{Cob}^{2+1}_n\to Cob^{2+1}_n.NEWLINE\]NEWLINE Introducing some suitable relations on \({\mathcal K}_n,{\mathcal S}_n\), and \({\mathcal H}^r_n\), the authors construct the corresponding quotient functors NEWLINE\[NEWLINE{\mathcal X}_n\to \bar{{\mathcal X}}_n\to \bar{\bar{{\mathcal X}}}_n,\;\;{\mathcal X}_n\to \bar{{\mathcal X}}_n^{c}\to \bar{\bar{{\mathcal X}}}_n^{c},NEWLINE\]NEWLINE where \({\mathcal X}={\mathcal K},{\mathcal S}\) or \({\mathcal H}^r\). It is shown that these categories are equivalent to each other. This proves Problem B.NEWLINENEWLINEApplying these results to branched coverings over \(B^4\), the authors show that any two \(n\)-labeled ribbon surfaces representing the 2-equivalent 4-dimensional 2-handlebodies are related to each other by a finite sequence of 4 types of labeled 1-isotopy moves and 2 types of ribbon moves. Interesting examples are the Akbulut-Kirby 4-balls \(\Delta_n\) [\textit{S. Akbulut} and \textit{R. Kirby}, Topology 24, 375--390 (1985; Zbl 0584.57009)]. It is well-known that any \(\Delta_n\) is diffeomorphic to the 4-ball [\textit{R. E. Gompf}, Topology 30, No. 1, 97--115 (1991; Zbl 0715.57016)] by introducing some 2/3-handle pair, in other words, 3-equivalent to the 4-ball. But it is not known whether \(\Delta_n\) is 2-equivalent to the 4-ball or not. Can we have a new invariant distinguishing them via those equivalent functors?NEWLINENEWLINEApplying these results to \(S^3\), the authors resolve Problem A and the graph version of the branched covering problem.