Universal Lefschetz fibrations and Lefschetz cobordisms (Q906843)

Let \(M\) and \(V\) be manifolds of dimensions respectively \(m+2\) and \(m+2k\) with \(m\geq 0\) and \(k\geq 2\). For \(f:V\to M\), the critical set of \(f\) is denoted by \(\widetilde A_f\) and the critical image of \(f\) by \(A_f=f(\widetilde A_f)\). A map \(f:V\to M\) is called a Lefschetz fibration if (i) near any critical point \(\tilde a\in \widetilde A_f\), \(f\) is locally equivalent to the map \(f_0:\mathbb R^m_+\times\mathbb C^k\to \mathbb R^m_+\times\mathbb C\) defined as \(f_0(x,z_1,\dots,z_k)=(x,z^2_1+\dots+z_k^2)\), where \(\mathbb R^m_+=\{(x_1,\dots,x_m);\;x_m\geq 0\}\), (ii) \(f_|:\widetilde A_f\to M\) is an embedding, and (iii) \(f_|:V-f^{-1}(A_f)\to M-A_f\) is a locally trivial bundle with fiber a manifold \(F\). For the oriented surface \(F=F_{g,b}\) of genus \(g\) with \(b\) boundary components let \(\mathcal{L}(F)\) be some class of Lefschetz fibrations with fiber \(F\). A Lefschetz fibration \(u:U\to M\) with fiber \(F\) is \(\mathcal{L}(F)\)-universal if (i) for any \(f:V\to N\) that belongs to \(\mathcal{L}(F)\) there exists a \(u\)-regular map \(q:N\to M\) such that \(q^*(u)\cong f\), and (ii) any such pullback for an arbitrary \(q:N\to M\) belongs to \(\mathcal{L}(F)\) up to equivalence, where an \(n\)-manifold \(N\) is the base of a Lefschetz fibration of \(\mathcal{L}(F)\). If \(\mathcal{M}_{g,b}\) is the mapping class group of \(F_{g,b}\), or the group of self-diffeomorphisms of \(F_{g,b}\) which keep \(\partial F_{g,b}\) fixed pointwise, and \(\widehat{\mathcal{M}}_{g,b}\) is the general mapping class group of \(F_{g,b}\), whose elements are the isotopy classes of orientation-preserving self-diffeomorphisms of \(F_{g,b}\), then the regular bundle associated with \(f:V\to M\) has a monodromy homomorphism \(\widehat\omega_f:\pi_1(M-A_f)\to\widehat{\mathcal{M}}_{g,b}\), called the bundle monodromy of \(f\). A homomorphism \(\omega^{\bowtie}_f:\pi_1(A_f)\to \widehat{\mathcal{M}}_{g-1,b,2}\), where \(\widehat{\mathcal{M}}_{g,b,}\) denotes the general mapping class group of \(F_{g,b}\) with \(n\) marked points, is called the singular monodromy of the singular bundle associated with \(f\). A canonical homomorphism \(\omega_f:\mu_1(N(A_f),A_f)\to\mathcal{M}_{g,b}\) is called the Lefschetz monodromy of \(f\), where \(\mu_1(M,A)\) is the subgroup of \(\pi_1(M-A)\) and \(N(A)\) is a compact tubular neighborhood of \(A\) in \(M\). A homomorphism \(\omega^{s}_f:\pi_2(M-A_f)\to \Pi_1(F)\), where \(\Pi(F)=\pi_1(\mathrm{Diff}(F),\mathrm{id})\), is called the structure monodromy of \(f\). Let \(\mathcal{C}_{g,b}\) denote the finite set of equivalence classes of homologically essential curves in \(F=F_{g,b}\) up to orientation-preserving diffeomorphisms of \(F\). If \(b\in\{0,1\}\), then \(\#\mathcal{C}_{g,b}=1\). In [Algebr. Geom. Topol. 12, No. 3, 1811--1829 (2012; Zbl 1270.57064)], the present author showed that a Lefschetz fibration \(u:U\to S\) over a surface with regular fiber \(F\) is universal with respect to bounded base surfaces if and only if \(\omega_u\) and \(\widehat \omega_u\) are surjective and any class of \(\mathcal{C}_{g,b}\) can be represented by a vanishing cycle of \(u\). In this paper, the author characterizes universal Lefschetz fibrations in dimension two by showing that a Lefschetz fibration \(u:U\to M\) with regular fiber \(F\) is universal with respect to the class of Lefschetz fibrations over a surface and with fiber \(F\), if (i) \(\widehat\omega_u\) is an isomorphism, (ii) \(\omega_u\) and \(\omega^s_u\) are surjective, and (iii) any class of \(\mathcal{C}_{g,b}\) can be represented by a vanishing cycle of \(u\). On the other hand, as a partial converse, \(u\) being universal implies (ii), (iii), and the surjectivity of \(\widehat\omega_u\). In particular, for \(g\geq 2\) and \(b\in\{0,1\}\), \(u\) is universal if \(\widehat\omega_u\) is an isomorphism and \(\omega_u\) is surjective. For dimension three, the author shows that if \(u:U\to M\) is a Lefschetz fibration with fiber \(F\) satisfying the following conditions: (i) \(\widehat\omega_u\) and \(\omega^s_u\) are isomorphisms, (ii) \(\omega_u\) and \(\omega^{\bowtie}_u\) are surjective, (iii) any class of \(\mathcal{C}_{g,b}\) can be represented by a vanishing cycle of \(u\), and (iv) \(A_u\) is connected, then \(u\) is universal for Lefschetz fibrations over \(3\)-manifolds and with fiber \(F\). Finally, the author presents an explicit construction of these fibrations and an application to Lefschetz cobordism groups, proving that these groups are quotients of certain singular bordism groups in dimension two and three. For the entire collection see [Zbl 1333.57003].