[Paper Review] Nielsen equivalence and trisections of 4-manifolds

Gabriel Islambouli|arXiv (Cornell University)|Apr 19, 2018

Geometric and Algebraic Topology11 references3 citations

TL;DR

This paper constructs distinct trisections of the same genus on fixed 4-manifolds using Nielsen equivalence of fundamental group generators from trisection spines. By applying techniques from Heegaard splitting theory—specifically Nielsen classes and automorphism actions—it proves the existence of $2^k - 1$ non-diffeomorphic $(3k,k)$-trisections on spun Seifert fiber spaces for each $k \geq 2$, demonstrating that stabilization is necessary to make non-isotopic trisections equivalent.

ABSTRACT

The goal of this paper is to construct distinct trisections of the same genus on a fixed 4-manifold. For every $k \geq 2$, we construct $2^{k}-1$ non-diffeomorphic $(3k,k)$-trisections on infinitely many 4-manifolds. Here, the manifolds are spun Seifert fiber spaces and the trisections come from Meier's spun trisections. The technique used to distinguish the trisections parallels an established technique for distinguishing Heegaard splittings. In particular, we show that the Nielsen classes of the generators of the fundamental group, obtained from spines of the 4-dimensional 1-handlebodies of the trisection, are isotopy invariants of the trisection. If we additionally consider the action of the automorphism group on the Nielsen classes, we obtain diffeomorphism invariants of trisections.

Motivation & Objective

• To construct multiple non-isotopic trisections of the same genus on a fixed 4-manifold.
• To demonstrate that stabilization is necessary to make distinct trisections isotopic, addressing a gap in trisection theory.
• To extend techniques from 3-manifold Heegaard splitting invariants—specifically Nielsen classes—to 4-dimensional trisections.
• To classify trisections up to diffeomorphism using automorphism-invariant Nielsen classes of fundamental group generators.
• To analyze small Seifert fiber spaces and show that some non-isotopic trisections become isotopic only after balanced stabilization, not unbalanced.

Proposed method

• Construct trisections via Meier’s spun trisection construction from Heegaard splittings of 3-manifolds.
• Associate to each trisection three Nielsen classes of generators from the spines of the 1-handlebodies in the trisection.
• Use Nielsen equivalence in the fundamental group to define isotopy invariants of trisections.
• Apply automorphism group actions on Nielsen classes to obtain diffeomorphism invariants of trisections.
• Leverage results from Lustig, Moriah, and Rosenberger on Nielsen equivalence in Fuchsian groups to classify generating sets.
• Analyze spun small Seifert fiber spaces with three exceptional fibers, using their known Heegaard splitting classification to construct trisections.

Experimental results

Research questions

• RQ1 Can multiple non-isotopic trisections of the same genus exist on a single 4-manifold?
• RQ2 Are there trisections that become isotopic only after stabilization, and if so, under what conditions?
• RQ3 Can Nielsen classes of fundamental group generators serve as invariants for trisection isotopy and diffeomorphism?
• RQ4 Is stabilization necessary to make non-isotopic trisections equivalent, and does the type of stabilization (balanced vs. unbalanced) affect this?
• RQ5 How do the Nielsen classes of spines in trisections relate to the underlying Heegaard splittings of the 3-manifold?

Key findings

• For every $k \geq 2$, there exist infinitely many 4-manifolds admitting $2^k - 1$ non-diffeomorphic $(3k,k)$-trisections.
• The Nielsen classes $\mathscr{N}(X_1), \mathscr{N}(X_2), \mathscr{N}(X_3)$ associated with the 1-handlebody spines are isotopy invariants of the trisection.
• If $\phi(\mathscr{N}(H_1)) \neq \mathscr{N}(H_1')$ for all $\phi \in \text{Aut}(\pi_1(M))$, then the corresponding spun trisections are not diffeomorphic.
• Three non-isotopic trisections on a fixed spun small Seifert fiber space become pairwise isotopic after a single balanced stabilization.
• These same trisections remain pairwise non-isotopic under any sequence of unbalanced $k$- and $l$-stabilizations, proving that balanced stabilization is essential.
• The trisections constructed are minimal genus and non-isotopic, showing that stabilization is genuinely required to achieve isotopy between distinct trisections.