[Paper Review] On diagrams of simplified trisections and mapping class groups

Kenta Hayano|Osaka City University (Osaka City University)|Nov 8, 2017

Geometric and Algebraic Topology5 references3 citations

TL;DR

This paper establishes a necessary and sufficient condition for a 3-tuple of curve systems on a surface to represent a simplified trisection using mapping class group theory. It provides an algorithm to derive trisection diagrams from vanishing cycles of simplified broken Lefschetz fibrations, and proves that Meier's trisections of spun 4-manifolds and certain genus-1 fibrations on $S^4$, $S^1\times S^3\sharp S^2\times S^2$, and $L_n$ are diffeomorphic to simplified trisections.

ABSTRACT

A simplified trisection is a trisection map on a 4-manifold such that, in its critical value set, there is no double point and cusps only appear in triples on innermost fold circles. We give a necessary and sufficient condition for a 3-tuple of systems of simple closed curves in a surface to be a diagram of a simplified trisection in terms of mapping class groups. As an application of this criterion, we show that trisections of spun 4-manifolds due to Meier are diffeomorphic (as trisections) to simplified ones. Baykur and Saeki recently gave an algorithmic construction of a simplified trisection from a directed broken Lefschetz fibration. We also give an algorithm to obtain a diagram of a simplified trisection derived from their construction.

Motivation & Objective

• To characterize when a 3-tuple of curve systems on a surface corresponds to a simplified trisection using mapping class groups.
• To develop an algorithmic method to construct trisection diagrams from vanishing cycles of simplified broken Lefschetz fibrations.
• To prove that trisections of spun 4-manifolds constructed by Meier are diffeomorphic to simplified trisections.
• To classify simplified trisections of genus 2 via linear algebraic reduction, avoiding deep results on Heegaard splittings.
• To verify that specific $(3,1)$-trisections on $S^4$ and related manifolds are equivalent to standard trisections via handle-slide sequences.

Proposed method

• Define a mapping class group condition on curve systems to determine if they arise from a simplified trisection, using vanishing cycles of indefinite folds.
• Use the theory of monodromy and parallel transport along loops between nested critical value circles in the trisection's critical set.
• Apply results from prior work on the effect of R2-moves on parallel transport to track changes in vanishing cycles during homotopy of stable maps.
• Construct trisection diagrams by translating vanishing cycles into curves via handle-slides and surgery, ensuring disjointness conditions.
• Use the mapping class group action to verify equivalence of trisection diagrams through explicit sequences of handle-slides.
• Apply the algorithm to genus-1 simplified broken Lefschetz fibrations to derive diagrams for $S^4$, $S^1\times S^3\sharp S^2\times S^2$, and $L_n$ manifolds.

Experimental results

Research questions

• RQ1 What is the necessary and sufficient condition in terms of mapping class groups for a 3-tuple of curve systems to represent a simplified trisection?
• RQ2 How can one algorithmically reconstruct a trisection diagram from the vanishing cycles of a simplified broken Lefschetz fibration?
• RQ3 Are the trisections of spun 4-manifolds constructed by Meier diffeomorphic to simplified trisections?
• RQ4 Can the classification of genus-2 simplified trisections be reduced to linear algebraic problems?
• RQ5 Are the $(3,1)$-trisections on $S^4$ obtained from genus-1 fibrations equivalent to the standard stabilized trisection?

Key findings

• A 3-tuple of curve systems on a genus-3 surface is a diagram of a simplified trisection if and only if the associated mapping class group condition on vanishing cycles is satisfied, as formalized in Theorem 3.4 and Lemma 3.5.
• The trisections of spun 4-manifolds due to Meier are diffeomorphic to simplified trisections, as proven in Theorem 3.7.
• The classification of simplified trisections of genus 2 is reduced to linear algebraic problems over the mapping class group, yielding a complete classification without relying on deep results on genus-2 Heegaard splittings of $S^3$.
• An algorithm is constructed to derive trisection diagrams from vanishing cycles of simplified broken Lefschetz fibrations, using monodromy and handle-slide operations.
• The $(3,1)$-trisection of $S^4$ obtained from a genus-1 simplified broken Lefschetz fibration is shown to be diffeomorphic to the stabilization of the $(0,0)$-trisection via explicit handle-slide sequences, as verified in Figure 20.
• The $(3,1)$-trisections on $L_0 \cong S^1\times S^3\sharp S^2\times S^2$ and $L_0' \cong S^1\times S^3\sharp S^2\tilde{\times}S^2$ are diffeomorphic to the connected sum of the $(1,1)$-trisection of $S^1\times S^3$ and the $(2,0)$-trisections of $S^2$-bundles over $S^2$.