Small Lefschetz fibrations and exotic 4-manifolds (Q522632)

A Lefschetz fibration on a closed, smooth, oriented \(4\)-manifold \(X\) is a smooth surjective map \(f:X\to S^2\) whose critical locus consists of finitely many points \(p_i\), such that at \(p_i\) and at \(f(p_i)\) there are local complex coordinates with respect to which \(f\) takes the form \((z_1,z_2)\mapsto z_1z_2\). The pair \((X,f)\) is called a genus-\(g\) Lefschetz fibration for the genus \(g\) of a regular fiber \(F\) of \(f\). A common way to construct new Lefschetz fibrations is the fiber sum operation. If \((X_i,f_i)\) is a genus-\(g\) Lefschetz fibration with regular fiber \(F_i\) for \(i=1,2\), then the fiber sum is a genus-\(g\) Lefschetz fibration \(f\) on \(X=(X_1,F_1)\#\varphi(X_2,F_2)\), obtained by removing a fibered tubular neighborhood of each \(F_i\) and then identifying the resulting boundaries via complex conjugation on \(S^1\) times a chosen orientation preserving diffeomorphism \(\varphi:F_1\to F_2\). A Lefschetz fibration \((X,f)\) is said to be indecomposable if it cannot be expressed as a fiber sum. A Lefschetz fibration \((X,f)\) is called minimal if there are no exceptional spheres contained in the fibers. If \(\Sigma^m_g\) is a compact, connected, oriented surface of genus \(g\) with \(m\) boundary components, then the group \(\Gamma_g^m\) composed of orientation-preserving self-homeomorphisms of \(\Sigma^m_g\) which restrict to the identity along \(\partial\Sigma^m_g\), modulo isotopies that restrict to the identity along \(\partial\Sigma^m_g\) as well, is called the mapping class group. If \(t_c\in\Gamma^m_g\) is the positive Dehn twist along the simple closed curve \(c\subset\Sigma^m_g\), \(\{c_i\}\) is a nonempty collection of simple closed curves on \(\Sigma^m_g\), \(\{\delta_j\}\) is a collection of \(m\) curves parallel to distinct boundary components of \(\Sigma^m_g\), \(\{k_j\}\) is a collection of \(m\) integers, and the relation \(t_{c_l}\cdots t_{c_2}t_{c_1}=t^{k_1}_{\delta_1}\cdots t^{k_m}_{\delta_m}\) holds in \(\Gamma^m_g\), then \(t_{c_l}\cdots t_{c_2}t_{c_1}\) is called a positive factorization of length \(l\) of the mapping class \(t^{k_1}_{\delta_1}\cdots t^{k_m}_{\delta_m}\) in \(\Gamma^m_g\). Existence of minimal symplectic structures on 4-manifolds is a fundamental question in smooth 4-manifold topology. There has been much interest in producing minimal symplectic 4-manifolds in the homeomorphism classes of standard simply-connected 4-manifolds with small second homology, such as blow-ups of \({\mathbb {CP}}^{2}\) or \(3 {\mathbb {CP}}^{2}\). In this paper, the authors demonstrate ways to construct positive factorization for Lefschetz fibrations with small number of critical points, as they correspond to 4-manifolds with small second homology which allow to provide simple descriptions of many new small exotic 4-manifolds. The authors show that there exist decomposable genus-2 Lefschetz fibrations whose total spaces are minimal symplectic 4-manifolds homeomorphic but not diffeomorphic to complex rational surfaces \({\mathbb {CP}}^{2}\# p\, \overline{\mathbb {CP}}^{2}\) for \(p=7, 8, 9\), and to \(3 {\mathbb {CP}}^{2}\#q\, \overline{\mathbb {CP}}^{2}\) for \(q =12,\dotsc,19\), and describe all the genus-2 Lefschetz fibrations explicitly via positive factorizations in the mapping class group of a genus-2 surface with one boundary component. They also show that any simply-connected minimal genus-2 Lefschetz fibration \((X,f)\) with \(b^+(X)\leq 3\) is homeomorphic to \({\mathbb {CP}}^{2}\# p\, \overline{\mathbb {CP}}^{2}\) for some \(7\leq p\leq 9\), or to \(3 {\mathbb {CP}}^{2}\#q\, \overline{\mathbb {CP}}^{2}\) for \(12\leq q\leq 19\). Finally, they prove that there exist decomposable minimal genus-2 Lefschetz fibrations over \(T^2\) and \(\sigma_2\) which are equivalent via Luttinger surgeries to minimal symplectic 4-manifolds \({\mathbb {CP}}^{2}\# 4 \overline{\mathbb {CP}}^{2}\) and \(3 {\mathbb {CP}}^{2}\# 6 \overline{\mathbb {CP}}^{2}\), respectively.