[Paper Review] Broken Lefschetz fibrations and mapping class groups

R. İnanç Baykur, Kenta Hayano|arXiv (Cornell University)|Oct 21, 2014

Geometric and Algebraic Topology8 references3 citations

TL;DR

This paper establishes a complete combinatorial classification of simplified broken Lefschetz fibrations (BLFs) on 4-manifolds via Hurwitz cycle systems, proving a bijection between isomorphism classes of BLFs and Hurwitz equivalence classes of such systems. The key contribution is the construction of infinitely many pairwise non-isomorphic BLFs on the same 4-manifold with isotopic regular fibers, demonstrating that BLF isomorphism is strictly finer than homotopy equivalence of generic maps.

ABSTRACT

The purpose of this note is to explain a combinatorial description of closed smooth oriented 4-manifolds in terms of positive Dehn twist factorizations of surface mapping classes, and further explore these connections. This is obtained via monodromy representations of simplified broken Lefschetz fibrations on 4-manifolds, for which we provide an extension of Hurwitz moves that allows us to uniquely determine the isomorphism class of a broken Lefschetz fibration. We furthermore discuss broken Lefschetz fibrations whose monodromies are contained in special subgroups of the mapping class group; namely, the hyperelliptic mapping class group and in the Torelli group, respectively, and present various results on them which extend or contrast with those known to hold for honest Lefschetz fibrations. Lastly, we show that there are 4-manifolds admitting infinitely many pairwise nonisomorphic relatively minimal broken Lefschetz fibrations with isotopic regular fibers.

Motivation & Objective

• To provide a complete combinatorial description of simplified broken Lefschetz fibrations (BLFs) on smooth 4-manifolds using mapping class group factorizations.
• To extend Hurwitz moves to BLFs, enabling unique determination of isomorphism classes via Hurwitz equivalence of cycle systems.
• To investigate BLFs with monodromy in special subgroups of the mapping class group, particularly the hyperelliptic and Torelli groups.
• To demonstrate that isomorphism classes of BLFs are strictly finer than homotopy classes of indefinite generic maps, by constructing infinite families of non-isomorphic fibrations with isotopic fibers.

Proposed method

• The authors define a Hurwitz cycle system (c; c₁,…,cₙ) as an ordered tuple of simple closed curves on a genus-g surface, encoding the monodromy of a BLF.
• They introduce two moves—elementary transformations and simultaneous conjugations—defining Hurwitz equivalence to classify cycle systems up to isomorphism of the corresponding BLFs.
• The main result establishes a bijection between isomorphism classes of genus-g BLFs and Hurwitz equivalence classes of such cycle systems for g ≥ 3.
• The paper uses the kernel of the map Φ_c: Map(Σ_g) → Map(Σ_{g-1}) to characterize valid monodromy factorizations.
• A geometric invariant I(d₁,d₂) is defined as the geometric intersection number of paths in Σ_c, which is preserved under Hurwitz moves and used to distinguish non-isomorphic fibrations.
• The construction of infinite non-isomorphic fibrations relies on paths with increasing intersection numbers, leading to distinct monodromy factorizations despite isotopic fibers.

Experimental results

Research questions

• RQ1 Can simplified broken Lefschetz fibrations on 4-manifolds be completely classified via combinatorial data from mapping class group factorizations?
• RQ2 How do Hurwitz moves extend to BLFs, and can they uniquely determine isomorphism classes of fibrations?
• RQ3 What invariants distinguish non-isomorphic BLFs with isotopic regular fibers?
• RQ4 Do fibrations with monodromy in the hyperelliptic or Torelli subgroups of the mapping class group exhibit properties analogous to or distinct from those of genuine Lefschetz fibrations?
• RQ5 Is the isomorphism class of a BLF strictly finer than the homotopy class of its underlying map to S²?

Key findings

• There exists a bijection between isomorphism classes of genus-g broken Lefschetz fibrations and Hurwitz equivalence classes of Hurwitz cycle systems for g ≥ 3.
• The geometric intersection number I(d₁,d₂) is invariant under Hurwitz moves and serves as a complete invariant for distinguishing non-isomorphic fibrations with monodromy in the kernel of Φ_c.
• An infinite family of pairwise non-isomorphic BLFs on the same 4-manifold X₀ is constructed, all having isotopic regular fibers.
• The fibrations fₙ and fₘ (n ≠ m) are non-isomorphic due to differing intersection numbers I(d₁,d₂ⁿ) = 4n + 2, which are preserved under Hurwitz equivalence.
• Despite non-isomorphism, the fibrations hₙ = fₙ ∘ Φ₀,ₙ are homotopic to each other and represent the same element in the cohomotopy set π²(X₀) = [X₀, S²].
• The construction shows that BLF isomorphism is strictly finer than homotopy equivalence of indefinite generic maps, highlighting a rigidity in the classification of 4-manifold fibrations.