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**Trisecting 4-manifolds.**
*(English)*
Zbl 1372.57033

This paper introduces trisections, a new decomposition of smooth 4-manifolds analogous to Heegaard splittings of 3-manifolds. The paper establishes fundamental properties and examples of trisections, with a particular emphasis on placing them within a larger framework.

A \((g,k)\)-trisection of a smooth 4-manifold \(X\) is a decomposition \(X=X_1\cup X_2\cup X_3\) into three genus \(k\) 1-handlebodies (i.e., \(X_i\cong \natural^k S^1\times D^3\)), glued along their boundaries which are each equipped with a fixed genus \(g\geq k\) Heegaard splitting. Namely, the triple intersection \(X_1\cap X_2\cap X_3\) is a closed orientable surface of genus \(g\) (the trisection surface) that separates \(\partial X_i\) into two 3-dimensional 1-handlebodies, and these are precisely the pairwise intersections \(X_i\cap X_{i+1}\) and \(X_i\cap X_{i-1}\) (indices modulo 3). A trisection is shown to arise from a smooth generic map \(X\to {\mathbb{R}}^2\) (a Morse 2-function) and a trisection of \({\mathbb{R}}^2\), in much the same way as a Heegaard splitting of a 3-manifold \(Y\) arises from a Morse function \(Y\to {\mathbb{R}}\) and a bisection of \({\mathbb{R}}\).

The main result of the paper is that every smooth, closed, orientable, connected 4-manifold admits a \((g,k)\)-trisection for some \(g,k\) (existence), and any two trisections for the same 4-manifold become isotopic after some iterations of a certain stabilization operation (uniqueness), akin to the Reidemeister-Singer theorem in dimension three. Extending the analogy, a trisection is represented by a trisection diagram, a particular triple of closed curves on a closed orientable surface, and the existence and uniqueness statements are formulated in terms of trisection diagrams and analogues of the usual operations on Heegaard diagrams. (In particular, trisection stabilization means replacing a trisection diagram by its connected sum with the first non-trivial trisection diagram of \(S^4\).) The authors, moreover, show how to recover a Kirby diagram from a trisection diagram.

Besides proving these results, the authors make a decided effort to relate trisections to other settings; for example, the existence result is proved in two independent ways, one “natural from the point of view of Morse 2-functions” and one using handlebody decompositions “in the spirit of [P. Ozsváth and Z. Szabó, Adv. Math. 202, No. 2, 326–400 (2006; Zbl 1099.53058)]”. Our summary of the paper’s contents below, however, is mainly limited to describing the relevant methods involved.

Sections 1 and 2 lay the foundations for working with trisections, and include an extended set of detailed examples. (For readers familiar with the terminology, notable among these is the authors’ observation that, with respect to the moment map \({{\mathbb{C}}}P^2\to {\mathbb{R}}^2\) for the standard torus action on \({{\mathbb{C}}}P^2\), the preimages of the trisected plane form a \((1,0)\)-trisection.) The authors propose several applications of the main result, including some 4-manifold invariants (but which, to the reviewer’s knowledge, are still not known to yield more than homological data).

Section 3 presents the first proof of existence, drawing largely on techniques developed by the authors in [Geom. Topol. 19, No. 5, 2465–2534 (2015; Zbl 1328.57019)] and which roughly goes as follows. The set of critical values of a Morse 2-function \(F:X\to {\mathbb{R}}^2\) consists of immersed curves in the plane (cusps and folds), crossing which changes the genus of the (surface) fibers by one, and the authors show that if these curves (weighted in relation to this genus change) lie in a particular form with respect to the trisected plane, then the obvious three preimages under \(F\) yield a trisection of \(X\). Such a Morse 2-function is constructed, modulo a series of intricate homotopies, by extending a Morse function \(f:X\to {\mathbb{R}}\) (interpreting this copy of \({\mathbb{R}}\) as the horizontal axis) in the vertical direction by Morse functions on the 3-dimensional fibers \(\{f^{-1}(t)\}_t\).

Central to the second proof of existence (in Section 4) and the proof of uniqueness (in Section 5) is a correspondence the authors establish between trisections and handlebody decompositions, which is perhaps best captured by the following (roughly restated) observation made in Lemma 13. Given a handlebody decomposition for \(X\) (with an appropriate number of handles of each index), there is no difficulty in taking \(X_1\) (\(X_3\), respectively) of a trisection to be the union of the handles of index 0 and 1 (3 and 4, respectively). Now, if there is a Heegaard splitting \(\partial X_1 = H\cup_{\partial} H'\) of the boundary of \(X_1\) such that the framed attaching circles of the 2-handles of \(X\) lie only in \(H\) and (after isotopy) lie geometrically dual on the Heegaard surface \(\partial H\) to distinct compressing curves for \(H\), then the union of a collar of \(H\) in \(X\) (a 4-dimensional 1-handlebody) and the 2-handles is still a 1-handlebody (of lower genus) and may thus be taken as \(X_2\) in a trisection, with \(\partial H\) the trisection surface.

To prove existence (without using Morse 2-functions), then, the authors show that by stabilizing a Heegaard splitting of \(\partial X_1\) (as above) if necessary, the attaching link of the 2-handles of a handlebody decomposition can always be placed in this way (roughly: the link is pressed onto the Heegaard surface and self-intersections are resolved by stabilization). Uniqueness amounts to showing that this procedure, up to trisection stabilization and isotopy, is unaffected by performing handle moves of the handlebody decomposition, and is proved along similar lines, together with the observation that stabilizing the Heegaard splitting above means precisely to stabilize the resulting trisection (modulo adding canceling handle pairs).

In the final brief section, a relative version of trisections is formulated. Using the non-relative case, short proofs are given to show that a smooth, compact, orientable, connected, boundary-connected 4-manifold, with a fixed open book decomposition on its boundary, has a relative trisection which restricts on the boundary to give the open book, and that two relative trisections of the same 4-manifold which agree on the boundary are isotopic after stabilizations (which are defined as in the non-relative case and take place in the interior).

A \((g,k)\)-trisection of a smooth 4-manifold \(X\) is a decomposition \(X=X_1\cup X_2\cup X_3\) into three genus \(k\) 1-handlebodies (i.e., \(X_i\cong \natural^k S^1\times D^3\)), glued along their boundaries which are each equipped with a fixed genus \(g\geq k\) Heegaard splitting. Namely, the triple intersection \(X_1\cap X_2\cap X_3\) is a closed orientable surface of genus \(g\) (the trisection surface) that separates \(\partial X_i\) into two 3-dimensional 1-handlebodies, and these are precisely the pairwise intersections \(X_i\cap X_{i+1}\) and \(X_i\cap X_{i-1}\) (indices modulo 3). A trisection is shown to arise from a smooth generic map \(X\to {\mathbb{R}}^2\) (a Morse 2-function) and a trisection of \({\mathbb{R}}^2\), in much the same way as a Heegaard splitting of a 3-manifold \(Y\) arises from a Morse function \(Y\to {\mathbb{R}}\) and a bisection of \({\mathbb{R}}\).

The main result of the paper is that every smooth, closed, orientable, connected 4-manifold admits a \((g,k)\)-trisection for some \(g,k\) (existence), and any two trisections for the same 4-manifold become isotopic after some iterations of a certain stabilization operation (uniqueness), akin to the Reidemeister-Singer theorem in dimension three. Extending the analogy, a trisection is represented by a trisection diagram, a particular triple of closed curves on a closed orientable surface, and the existence and uniqueness statements are formulated in terms of trisection diagrams and analogues of the usual operations on Heegaard diagrams. (In particular, trisection stabilization means replacing a trisection diagram by its connected sum with the first non-trivial trisection diagram of \(S^4\).) The authors, moreover, show how to recover a Kirby diagram from a trisection diagram.

Besides proving these results, the authors make a decided effort to relate trisections to other settings; for example, the existence result is proved in two independent ways, one “natural from the point of view of Morse 2-functions” and one using handlebody decompositions “in the spirit of [P. Ozsváth and Z. Szabó, Adv. Math. 202, No. 2, 326–400 (2006; Zbl 1099.53058)]”. Our summary of the paper’s contents below, however, is mainly limited to describing the relevant methods involved.

Sections 1 and 2 lay the foundations for working with trisections, and include an extended set of detailed examples. (For readers familiar with the terminology, notable among these is the authors’ observation that, with respect to the moment map \({{\mathbb{C}}}P^2\to {\mathbb{R}}^2\) for the standard torus action on \({{\mathbb{C}}}P^2\), the preimages of the trisected plane form a \((1,0)\)-trisection.) The authors propose several applications of the main result, including some 4-manifold invariants (but which, to the reviewer’s knowledge, are still not known to yield more than homological data).

Section 3 presents the first proof of existence, drawing largely on techniques developed by the authors in [Geom. Topol. 19, No. 5, 2465–2534 (2015; Zbl 1328.57019)] and which roughly goes as follows. The set of critical values of a Morse 2-function \(F:X\to {\mathbb{R}}^2\) consists of immersed curves in the plane (cusps and folds), crossing which changes the genus of the (surface) fibers by one, and the authors show that if these curves (weighted in relation to this genus change) lie in a particular form with respect to the trisected plane, then the obvious three preimages under \(F\) yield a trisection of \(X\). Such a Morse 2-function is constructed, modulo a series of intricate homotopies, by extending a Morse function \(f:X\to {\mathbb{R}}\) (interpreting this copy of \({\mathbb{R}}\) as the horizontal axis) in the vertical direction by Morse functions on the 3-dimensional fibers \(\{f^{-1}(t)\}_t\).

Central to the second proof of existence (in Section 4) and the proof of uniqueness (in Section 5) is a correspondence the authors establish between trisections and handlebody decompositions, which is perhaps best captured by the following (roughly restated) observation made in Lemma 13. Given a handlebody decomposition for \(X\) (with an appropriate number of handles of each index), there is no difficulty in taking \(X_1\) (\(X_3\), respectively) of a trisection to be the union of the handles of index 0 and 1 (3 and 4, respectively). Now, if there is a Heegaard splitting \(\partial X_1 = H\cup_{\partial} H'\) of the boundary of \(X_1\) such that the framed attaching circles of the 2-handles of \(X\) lie only in \(H\) and (after isotopy) lie geometrically dual on the Heegaard surface \(\partial H\) to distinct compressing curves for \(H\), then the union of a collar of \(H\) in \(X\) (a 4-dimensional 1-handlebody) and the 2-handles is still a 1-handlebody (of lower genus) and may thus be taken as \(X_2\) in a trisection, with \(\partial H\) the trisection surface.

To prove existence (without using Morse 2-functions), then, the authors show that by stabilizing a Heegaard splitting of \(\partial X_1\) (as above) if necessary, the attaching link of the 2-handles of a handlebody decomposition can always be placed in this way (roughly: the link is pressed onto the Heegaard surface and self-intersections are resolved by stabilization). Uniqueness amounts to showing that this procedure, up to trisection stabilization and isotopy, is unaffected by performing handle moves of the handlebody decomposition, and is proved along similar lines, together with the observation that stabilizing the Heegaard splitting above means precisely to stabilize the resulting trisection (modulo adding canceling handle pairs).

In the final brief section, a relative version of trisections is formulated. Using the non-relative case, short proofs are given to show that a smooth, compact, orientable, connected, boundary-connected 4-manifold, with a fixed open book decomposition on its boundary, has a relative trisection which restricts on the boundary to give the open book, and that two relative trisections of the same 4-manifold which agree on the boundary are isotopic after stabilizations (which are defined as in the non-relative case and take place in the interior).

Reviewer: Ash Lightfoot (Moskow)