Completing the group-trisection dictionary for smooth 4-manifolds: the homeomorphism equivalence relation, group-theoretic reductions of the smooth 4-dimensional Poincaré conjecture, and trisection genus versus exotic smooth structures

Abrams, Gay and Kirby (2018) established a dictionary encoding every closed, connected, oriented, smooth 4-manifold, up to diffeomorphism, as a group trisection: a commutative cube of finitely presented groups in which every homomorphism is surjective and every face is a pushout square, taken up to isomorphism and an algebraic stabilization operation. This makes smooth 4-manifold topology, in principle, a branch of combinatorial group theory, and it restates the smooth 4-dimensional Poincaré conjecture (SPC4) as the assertion that every $(3k,k)$-trisection of the trivial group is stably equivalent to the trivial trisection. The task is to make progress on the three linked open problems that the source poses or suggests for the dictionary's algebraic side: (1) characterize, in purely algebraic terms, the equivalence relation on group trisections corresponding to homeomorphism rather than diffeomorphism of the underlying 4-manifolds — the level at which exotic smooth structures appear; (2) settle the group-theoretic reformulation of SPC4 stated in AGK's Corollary 6; and (3) formulate precisely, and settle, the expectation that unbounded trisection genus is necessary to realize all countably many exotic smooth structures on a fixed simply connected topological 4-manifold such as the K3 surface. Partial results are known in the standard direction: genus-two trisections are standard (Meier–Zupan) and weakly reducible genus-three trisections are standard (Aranda–Zupan).

Background

In 2012–2016, Gay and Kirby introduced trisections: every closed, connected, oriented, smooth 4-manifold $X$ admits a decomposition $X=V_1\cup V_2\cup V_3$ into three 4-dimensional handlebodies $V_i\cong\natural^k(S^1\times B^3)$ (boundary-connected sums of $S^1\times B^3$), glued so that each pairwise intersection $V_i\cap V_j$ is a 3-dimensional handlebody $\natural^g(S^1\times D^2)$ and the triple intersection $\Sigma_g=V_1\cap V_2\cap V_3$ is a closed orientable surface of genus $g$. Such a decomposition is a $(g,k)$-trisection, and the minimal genus $g$ over all trisections of $X$ is the trisection genus, a smooth invariant of $X$ analogous to the Heegaard genus of a 3-manifold (a Heegaard splitting decomposes a 3-manifold into two handlebodies; the trisection is its 4-dimensional analogue with three pieces). The Euler characteristic satisfies $\chi(X)=2+g-3k$, and any two trisections of the same manifold become isotopic after finitely many stabilizations (a local move increasing the genus $g$ by 3 and $k$ by 1), the analogue of the Reidemeister–Singer theorem for Heegaard splittings.

Abrams, Gay and Kirby (AGK) then gave a purely group-theoretic encoding of this picture. A $(g,k)$-trisection of a finitely presented group $G$ is a commutative cube of groups and surjective homomorphisms in which the bottom vertex carries the surface group $S_g=\pi_1(\Sigma_g)$, the three adjacent vertices carry free groups $H_g\cong F_g$ (the fundamental groups of the pairwise-intersection handlebodies), the next three vertices carry free groups of rank $k$ (the fundamental groups of the sectors $V_i$), and every face of the cube is a pushout square (a free product with amalgamation): each sector group is an amalgam of two handlebody groups over the surface group, and $G$ itself is recovered as an iterated amalgamated free product of the sector groups over the handlebody groups. AGK's main theorem gives a bijection between $(g,k)$-trisections of $G$ up to isomorphism and trisections of 4-manifolds with fundamental group $G$ up to trisection-preserving diffeomorphism (a diffeomorphism respecting the decomposition), compatible with an algebraic stabilization operation on both sides (on the group side, stabilization is the connected sum of the given group trisection with the standard trivial $(3,1)$-trisection of the trivial group, which turns a $(g,k)$-trisection of $G$ into a $(g+3,k+1)$-trisection of the same group $G$ — the algebraic counterpart of the geometric stabilization); after quotienting by stabilization on both sides, the bijection identifies group trisections of $G$ with diffeomorphism classes of smooth closed oriented 4-manifolds with fundamental group $G$. In consequence, smooth 4-manifold topology modulo diffeomorphism is, in principle, entirely group theory.

The dictionary is stated at the level of diffeomorphism, and this is exactly where its open problems lie, because smooth 4-manifold topology differs sharply from its higher-dimensional siblings: many closed simply connected topological 4-manifolds — the K3 surface, a simply connected complex surface that is the standard example — carry countably infinitely many pairwise non-diffeomorphic smooth structures, called exotic smooth structures, so homeomorphism (an equivalence of the underlying topological manifolds) is strictly coarser than diffeomorphism. AGK pose three interlinked problems. First, since their bijection captures diffeomorphism, they ask for an algebraic understanding of the equivalence relation on group trisections corresponding to homeomorphism between 4-manifolds. Second, the smooth 4-dimensional Poincaré conjecture (SPC4) — the assertion that every smooth homotopy 4-sphere, i.e., a smooth closed 4-manifold homotopy equivalent to $S^4$, is diffeomorphic to $S^4$ — remains one of the central open problems of topology, and AGK's Corollary 6 restates it algebraically: SPC4 holds if and only if every $(3k,k)$-trisection of the trivial group is stably equivalent to the trivial trisection (the $(0,0)$-trisection, which corresponds to $S^4$); indeed, a $(3k,k)$-trisection of the trivial group has $\chi=2$ and trivial fundamental group, hence yields a homotopy $S^4$. AGK propose two reductions modeled on Waldhausen's theorem from 3-manifold topology (every Heegaard splitting of $S^3$ is a stabilization of the genus-zero splitting): (a) prove that there is a unique $(3,1)$-trisection of the trivial group up to isomorphism, or at least that every $(3,1)$-trisection of the trivial group gives a 4-manifold diffeomorphic to $S^4$; and (b) prove that for any $(3k,k)$-trisection of the trivial group, the intersection of the three kernels $\ker(S_g\to H_g)$ of the edge homomorphisms contains a nontrivial element that can be represented by an embedded closed curve in $\Sigma_g$, since such a curve would decompose the trisection as a connected sum of lower-genus trisections and enable an induction; AGK note that the full strategy would prove more than SPC4, namely a 4-dimensional Waldhausen theorem (every trisection of $S^4$ is a stabilization of the trivial one). Third, AGK observe that the exotic smooth structures on a given simply connected topological 4-manifold correspond to a subset of the trisections of its (trivial, for K3) fundamental group, and remark that capturing the full countable collection of exotic structures seems likely to require unbounded trisection genus $g$. The two forms of (a) have distinct failure modes: a non-standard trisection of $S^4$ — a $(3,1)$-trisection of the trivial group not isomorphic to the standard one whose underlying manifold is still diffeomorphic to $S^4$ — would refute the 4-dimensional Waldhausen property without settling SPC4, whereas a $(3,1)$-trisection of the trivial group whose underlying 4-manifold is not diffeomorphic to $S^4$ would be an exotic 4-sphere and would refute SPC4.

Problem Statement

The problem is to complete the algebraic side of the AGK group-trisection dictionary along the three interlinked directions the source paper poses or suggests. The common scientific objective is an effective, purely group-theoretic account of the homeomorphism-versus-diffeomorphism distinction for closed oriented smooth 4-manifolds inside the group-trisection correspondence, with the smooth 4-dimensional Poincaré conjecture (SPC4) as its flagship special case. Concretely:

Q1 (homeomorphism equivalence relation). Characterize, in terms of the groups and homomorphisms constituting group trisections, the equivalence relation on group trisections that corresponds to homeomorphism (rather than diffeomorphism) of the associated 4-manifolds; in particular, give a group-theoretic criterion for when two group trisections yield homeomorphic but non-diffeomorphic 4-manifolds.

Q2 (group-theoretic SPC4). Settle AGK's Corollary 6 — decide whether every $(3k,k)$-trisection of the trivial group is stably equivalent to the trivial trisection.

Q3 (genus growth versus exotic structures). Formulate precisely, and prove or refute, the expectation that realizing all countably many exotic smooth structures on a fixed simply connected closed topological 4-manifold (such as the K3 surface) requires trisections — equivalently, group trisections of its fundamental group — of unbounded genus; in particular, determine whether trisection genus is bounded along any infinite family of pairwise non-diffeomorphic smooth structures on the same topological manifold.

The three questions are jointly directed at one objective — an effective, purely group-theoretic account of the homeomorphism-versus-diffeomorphism distinction for closed oriented smooth 4-manifolds. Taken together they determine whether trisection genus is a faithful measure of smooth complexity rather than of topology.

References

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