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MATHEMATICS SPECIAL FEATURE respect to this symmetric presentation of H2(X ), the intersection 2. Three-Dimensional Lemmas form is given by the following equation. Our four-dimensional calculations will depend on our ability to compute intersections and linking numbers between various 0 0 0 0 (a, b, c), (a , b , c ) = ha, b iΣ. [2] attaching circles and belt spheres embedded in S 3 or #k (S 1 × S 2). The lemmas of this section describe how to carry out these In Section 4, we describe various methods of computing the calculations inside the first homology of a Heegaard splitting intersection form directly from a trisection diagram. In partic- surface in one of these 3-manifolds. ular, Propositions 4.2 and 4.3 give explicit formulas in terms of For a closed 3-manifold M we denote by h·, ·iM the intersec- the three algebraic intersection matrices arising from the defin- tion pairing on H2(M ) × H1(M ), and for a closed surface Σ we ing curves of the trisection diagram. The following example denote by h·, ·iΣ the intersection pairing on H1(Σ) × H1(Σ). demonstrates one of our methods. Example: Let us compute the intersection form of the 2 2 Lemma 2.1. Suppose W is a 3D handlebody with boundary surface (2; 0, 0, 0)-trisection of S × S with diagram shown in Fig. 1. Σ, and let ι∗ : H1(Σ) → H1(W ) be the map induced by inclusion. Using the αi , βi , γi as labeled in the figure, we obtain the Then: following intersection matrices. 1. If {α1, ... , αg } is any defining set of curves for W in Σ, then  0 1  0 −1  0 1 {[α1], ... ,[αg ]} forms a basis of ker(ι∗), and αQβ = −1 0 , γ Qβ = −1 0 , αQγ = −1 0 . 2. For all x, y ∈ ker(ι∗), hx, yiΣ = 0 and ker(ι∗) is maximal with respect to this property. In other words, ker(ι∗) is a maximal Lagrangian subspace of H1(Σ). Using Proposition 4.3, we find that the intersection form of S 2 × 2 ∼ 2g ∼ g S has matrix Proof: Since H1(Σ) = Z , H1(W ) = Z , and ι∗ is a sur- jection, ker(ι∗) is a free abelian subgroup of rank g. Since   −1 0 1 {α1, ... , αg } is a defining set of curves, the set {[α1], ... ,[αg ]} is Φ 2 2 = γ Qβ (αQβ ) αQγ = , S ×S 1 0 linearly independent in H1(Σ) and the subgroup it generates lies inside of ker(ι∗). This proves Claim (1). as expected. If x, y ∈ ker(ι∗), then by Claim (1) x and y can be repre- The strategy we use to compute the homology of X is to estab- sented by disjoint collections of curves isotopic to the elements lish that the complex in Eq. 1.1 is isomorphic to a chain complex of {α1, ... , αg }, and so it follows that hx, yiΣ = 0. Moreover, if z arising from a handle decomposition of the trisected 4-manifold is any element of H1(Σ) satisfying hx, ziΣ = 0 for all x ∈ ker(ι∗), then any representative ζ of z can be band-summed along sub- X . However, as an amusing exercise, one may use the fact that 0 any two trisection diagrams for X are related by handle slides arcs of the α-curves to a new representative ζ of z which is and stabilizations (4) to show that all chain complexes of the disjoint from the α-curves. This implies that z ∈ ker(ι∗), which form of Eq. 1.2 coming from trisections of X are chain homotopy establishes the claim that ker(ι∗) is maximal. Alternatively, max- equivalent. Similarly, one many directly check that for any two imality follows from the observation that the α-curves form half trisections of X the pairings described in Eq. 1.2 are isomorphic. of a symplectic basis of H1(Σ). Finally, Theorem 4.4 asserts that all (g; k, 0, 0)-trisections For the remainder of this section, let (Σ; Wα, Wβ ) be a admit “algebraically trivial” diagrams. One consequence of this Heegaard splitting of a 3-manifold M , let ιM : H1(Σ) → H1(M ) is that the triple of algebraic intersection matrices arising from and ι : H1(Σ) → H1(W) denote the maps induced by inclusion a trisection diagram do not contain any additional information (for  ∈ {α, β}), and set L = ker(ι). beyond the homology and intersection form (thus giving a nega- For the following lemma, it is necessary to have an understand- tive answer to a speculative question appearing in ref. 4, at least ing of the Mayer–Vietoris map ∂∗ : H2(M ) → H1(Σ). Suppose for this class of trisections). Q ⊂ M is an oriented embedded surface which represents x ∈ One may speculate on whether Theorem 4.4 has further appli- H2(M ) and is transverse to Σ. Then ∂∗x is represented by the cations. There is a large difference between algebraically trivial family of curves of Q ∩ Σ with the orientations they inherit as diagrams and geometrically trivial diagrams—a gap which is rem- the boundary of the oriented surface Q ∩ Wα using the standard iniscent of the failure of the h-cobordism theorem for cobordisms outward normal first convention. between 4-manifolds. On the other hand, if new restrictions on intersection forms are to be found using trisection diagrams (or Lemma 2.2. Let ∂∗ : H2(M ) → H1(Σ) be the boundary map com- if old, previously known restrictions are to be recovered more ing from the Mayer–Vietoris sequence for the triple (Σ; Wα, Wβ ). easily), then it seems likely that Theorem 4.4 will play a role. Then: Additionally, the result suggests that the relationship between 1. The map ∂∗ is an isomorphism between H2(M ) and Lα ∩ Lβ ; the Torelli group and the Georitz group of a surface might 2. For any z ∈ H2(M ) and y ∈ H1(Σ), hιM y, ziM = h∂∗z, yiΣ; play an important role in understanding exotic 4-dimensional 3. If H1(M ) is torsion free and x ∈ H1(Σ), then x ∈ ker(ιM ) if and phenomena. only if hy, xiΣ = 0 for all y ∈ Lα ∩ Lβ . Proof: In the following portion of the Mayer–Vietoris sequence

,

we have H2(Wα) ⊕ H2(Wβ ) = 0 and ker(ια ⊕ ιβ ) = Lα ∩ Lβ , so by exactness ∂∗ is an injective map onto Lα ∩ Lβ . For the second claim, choose a representative Q of z which is transverse to Σ, and a representative γ of y which is transverse to Q ∩ Σ. Then γ also represents ιM y, Q ∩ Σ represents ∂∗z, and since all intersections between γ and Q lie in Q ∩ Σ it follows 2 2 Fig. 1. A (2; 0, 0, 0)-trisection diagram for S × S . that hz, ιM yiM agrees with h∂∗z, yiΣ up to sign. The question of

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MATHEMATICS SPECIAL FEATURE represented by its belt sphere, and let a2 : C2 → Lγ be the isomor- Proof: First we prove that phism which sends each 2-handle generator of C2 to the element of Lγ ⊂ H1(Σ) that is represented by its attaching sphere (which Lγ ∩ (Lα + Lβ ) H2(X ) =∼ . [3.9] by construction is embedded in Σ). In this notation, Eq. 3.2 takes (L ∩ L ) + (L ∩ L ) the following form: γ α γ β

If ∂2 and ∂3 are the maps from the complex in Theorem 3.1, then ˆ X ∂2(h) = hb1(hj ), ι∂V1 a2(h)i∂V1 hj . [3.4] ker(∂2) consists of those elements of Lγ which are orthogonal j ⊥ ⊥ to Lα ∩ Lβ in H1(Σ). By Lemma 2.1, Lα = Lα and Lβ = Lβ , so ⊥ (Lα ∩ Lβ ) = Lα + Lβ . Hence, ker(∂2) = Lγ ∩ (Lα + Lβ ). Addi- The map ι∂V1 : H1(Σ) → H1(∂V1) is induced by inclusion. By tionally, one can readily see that Im(∂3) = (Lγ ∩ Lα) + (Lγ ∩ Lemma 2.2, this equation can in turn be replaced by Lβ ), so Eq. 3.9 now follows from Theorem 3.1. To prove the symmetric formula holds, first note that X ∂ˆ2(h) = h∂b1(hj ), a2(h)iΣ hj . [3.5] (a, b, c) 7→ c induces j {(a, b, c) ∈ Lα × Lβ × Lγ | a + b − c = 0} =∼ Lγ ∩ (Lα + Lβ ), Thus, if ∂∗ is the Mayer–Vietoris map of Lemma 2.2, D : Lα ∩ {a + b = 0} Lβ → Hom(Lα ∩ Lβ , Z) is the dual map induced by the basis ele- ments ∂∗b1(hj ), and we define f1 = D ◦ ∂∗ ◦ b1 and f2 = a2, then and so we obtain L ∩ (L + L ) ∂2 ◦ f2 = f1 ◦ ∂ˆ2. [3.6] γ α β (Lγ ∩ Lα) + (Lγ ∩ Lβ ) 2 3 2 By construction, C3 splits as C3 ⊕ C3 , where C3 is the free abelian {(a, b, c) ∈ Lα × Lβ × Lγ | a + b − c = 0} 3 =∼ . group generated by the 3-handles defining V2 and C3 is its {a + b = 0} + {a − c = 0} + {b − c = 0} analogue for V3. Thus, we obtain an isomorphism a3 : C3 → 2 H2(∂V2) ⊕ H2(∂V3) which maps each 3-handle generator of C3 3 3 The isomorphism H1(Σ) → H1(Σ) induced by (a, b, c) 7→ to the element of H2(∂V2) represented by its attaching sphere 3 (a, b, −c) induces an isomorphism with in ∂V2, and likewise for the generators of C3 and H2(∂V3). Let f3 : C3 → (Lα ∩ Lγ ) ⊕ (Lβ ∩ Lγ ) be the map obtained by compos- {(a, b, c) ∈ Lα × Lβ × Lγ | a + b + c = 0} ing a3 with the sum of the ∂∗ maps H2(∂V2) → Lα ∩ Lγ and , {a + b = 0} + {a + c = 0} + {b + c = 0} H2(∂V3) → Lβ ∩ Lγ from Lemma 2.2. We now prove that which is equal to the desired expression. ∂3 ◦ f3 = f2 ◦ ∂ˆ3. [3.7]

Corollary 3.4. H3(X ) =∼ Lα ∩ Lβ ∩ Lγ . This, together with Eq. 3.6, will show chain complex in (Eq. 3.3) is Proof: From Theorem 3.1 we have H3(X ) =∼ ker(∂3). Sup- isomorphic to (the middle part of) the complex in the statement pose that (x, y) ∈ (Lα ∩ Lγ ) ⊕ (Lβ ∩ Lγ ) and ∂3(x, y) = 0. Then of the theorem, which will complete the proof. x = −y and, since y ∈ Lβ ∩ Lγ , we have x ∈ Lβ . Thus, the Recall that we chose 2-handles for the handle decomposition map f : Lα ∩ Lβ ∩ Lγ → ker(∂3) given by f (x) = (x, −x) is an of X whose cores form a defining collection of disks for Wγ . isomorphism. Thus, every sphere S ⊂ ∂V2 (or ∂V3) is isotopic to a sphere Definition 3.5: Let Φ:(Lγ ∩ (Lα + Lβ )) × (Lγ ∩ (Lα + Lβ )) → which intersects Wγ only in disks that are parallel to the core Z be given by the formula disks of the 2-handles. (This follows from a standard innermost and outermost disk argument.) If a component E of S ∩ Wγ is 0 Φ(x, y) = hx , yiΣ, [3.10] isotopic in Wγ to the core of the 2-handle hj , then it is disjoint from the belt sphere of all other 2-handles and meets the belt 0 0 where x is any element of Lα which satisfies x − x ∈ Lβ . sphere of hj in a single intersection point. Moreover, this inter- 0 section is positive if the orientation of E matches that of the The element x above exists since x ∈ Lα + Lβ , and although it will not generally be unique, the value of Φ(x, y) does not core of hj , and negative otherwise. Thus, if h is a 3-handle of depend on the choice of x 0. The proof of this is formally identical the decomposition, then in Eq. 3.2 above τj counts (with sign) to the last paragraph of the proof of Lemma 2.4. the number of parallel copies of the core of hj that lie in S ∩ Wγ , where S is the attaching sphere of h. It follows that, for every Theorem 3.6. The group generator h ∈ C3, f2 ◦ ∂3(h) is the element of H1(Σ) represented by S ∩ Σ. The same description applies to ∂3 ◦ f3(h), so we have L ∩ (L + L ) shown that Eq. 3.7 holds. γ α β , The chain complex of Theorem 3.1 allows us to find presenta- (Lγ ∩ Lα) + (Lγ ∩ Lβ ) tion matrices for H1(X ), but aside from this it does not appear to yield any particularly elegant characterizations of H1(X ). How- together with the form induced by Φ is isomorphic to H2(X ) ever, we do obtain the following descriptions of H2(X ) and together with the intersection form. H3(X ) which are symmetric with respect to the α, β, and γ Proof: Define a handle decomposition of X as in the proof of 1 2 curves. Theorem 3.1, so that X = V1 and X is the union of V1 with a regular neighborhood of a defining collection of disks for Hγ ; we Corollary 3.3. Writing {a + b + c = 0} for {(a, b, c) ∈ Lα × Lβ × retain the notation for Ci and ∂i as well. 2 Lγ | a + b + c = 0}, we have There is an isomorphism f : H2(X ) → Lγ ∩ (Lα + Lβ ) which 2 factors through ker(∂2). Explicitly, given any z ∈ H2(X ) and an {a + b + c = 0} oriented surface Q representing it, let H2(X ) =∼ , [3.8] {c = 0} + {b = 0} + {a = 0} g ˜ X where the summands in the denominators are understood to be the f (z) = τj hj , [3.11] subgroups of {a + b + c = 0} satisfying the given linear equations. j =1

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MATHEMATICS SPECIAL FEATURE of Φ. Note that all 4-manifolds which admit handle decomposi- following paragraph we explain how the dij moves can be real- tions without 1-handles will admit such trisections by the above ized geometrically as well, in a way that maintains Conditions (1) construction. and (2). Specifically, the column move in dij can be geometrically real- Proposition 4.3. Suppose (α, β, γ) is a (g; 0, k2, k3)-trisection dia- ized by replacing αj with a curve obtained by sliding αi over gram. Then Φ is given on Lγ by the matrix αj , and similarly the row move corresponds to replacing γj with a curve obtained by sliding γi over γj . These handleslides will −1 Φ = γ Qβ (αQβ ) αQγ . change αQβ by adding its ith row to its j th row, and they will change γ Qβ by adding its ith column to its j th column. However, Proof: The proof is a computation that follows the pattern so long as we have chosen the arc defining our handleslide of αi of Proposition 4.2, but in this case Hα and Hβ form a Heegaard over αj to be disjoint from all of the α and β curves except at its 3 splitting of S . Thus, Lγ ∩ (Lα + Lβ ) = Lγ and β Qα is invertible, endpoints (as can always be done), then we can always perform 0 −1 β so we can solve Eq. 4.3 to obtain x = (β Qα) β Qγ x. For any a cancellation move which replaces the curve i with the result β −β x, y ∈ Lγ , we therefore have of sliding of i over j in the manner shown in Fig. 2. This geometric move makes the α and β curves satisfy Condition (1) −1 T T −1 again, and on the algebraic level it corresponds to a subtraction Φ(x, y) = ((β Qα) β Qγ x) αQγ y = x γ Qβ (αQβ ) αQγ y. of the j the column of αQβ from its ith column, and a subtraction This completes the proof. of the j th row of γ Qβ from its ith row. Thus, we can geometri- By ref. 6, every 4-manifold which admits a handle decomposi- cally realize the move dij while maintaining Conditions (1) and −d . tion without 1- or 3-handles admits a (g; 0, k2, 0)-trisection, and (2), and the same reasoning works for ij we can make the following statement about such trisections. It is tempting to hope that Theorem 4.4 can be used to prove the triviality of a large class of trisections. Specifically, if one

Theorem 4.4. Suppose T is a (g; 0, k2, 0)-trisection of X , and let could show that it is always possible to find a trisection diagram |ω ∩ σ| = |hω, σi | QX be any matrix representing the intersection form of X . Then T satisfying the equation Σ for all pairs of curves admits a diagram (α, β, γ) with the following properties: ω and σ appearing in the diagram, then the triviality or near- triviality (in the case when the form contains E8 summands) of 3 1. (α, β) is a standard diagram of S , with indices labeled so that all (g; 0, k2, 0)-trisections would follow by choosing an appropri- |αi ∩ βj | = δij ate QX in Theorem 4.4. However, a paper of Meier and Zupan 2. γ Qβ = Id appearing in this collection (10) gives examples of exotic con- 2 2 3. The matrix αQγ has QX as its upper left minor, and is zero nected sums of CP ’s and CP ’s admitting (g; 0) trisections. Such elsewhere. manifolds cannot admit geometrically trivial trisection diagrams, Proof: Given any diagram (α, β, γ) of T , by Waldhausen’s even though Theorem 4.4 tells us that such manifolds will admit theorem on the uniqueness of Heegaard splitting surfaces for homologically trivial diagrams. S 3 (9), we can perform handle slides of the α and β curves so Moreover, even when a trisection admits a trivial diagram, that Condition (1) is satisfied. Moreover, we can orient the com- other diagrams of the same trisection can be chosen in which the −1 discrepancy between geometric intersection numbers and alge- ponents of α and β so that αQβ = (αQβ ) = Id. Since γ Qβ is 3 braic intersections numbers is arbitrarily large. For example, the a presentation matrix of H1(S ) = 0 it is unimodular, so after 2 2 standard (2; 0) trisection of # does indeed admit a trivial a sequence of row operations it can be reduced to the identity CP CP diagram (α, β, γ) such that |αi ∩ βj | = |αi ∩ γj | = |βi ∩ γj | = δij matrix, and these row operations can be geometrically realized. for 1 ≤ i, j , ≤ 2. However, there exist separating curves σ1 and Specifically, the act of multiplying the ith row of γ Qβ by −1 σ2 on Σ which bound disks in both Wα and Wβ , and such that corresponds to reversing the orientation of γi , the act of switch- Σ \ (σ1 ∪ σ2) consists entirely of disks. Thus, if f :Σ → Σ is the ing row i with row j corresponds to swapping the indices of γi composition of the positive Dehn twist about σ1 followed by and γj , and the act of adding or subtracting row i from row j the negative Dehn twist about σ2, then f is a pseudo-Anosov corresponds to performing a handle slide of γi over γj and then map [by Penner’s construction (11)] lying in the intersection of replacing γj with the resulting curve. Hence, after a sequence of the mapping class group of (Σ; Wα, W ) and the Torelli sub- handle slides, orientation changes, and reindexing moves on the β group of Σ. If dΣ is simpicial distance in the curve complex, γ curves, Conditions (1) and (2) will both be satisfied by (α, β, γ). n it follows that dΣ(γi , f (γi )) goes to infinity as n does, which This being done, by Proposition 4.3 the matrix αQγ can be n in turn implies that the quantities |f (γi ) ∩ αi | go to infinity as converted to the form required by Condition (3) by a sequence well [as was shown by Hempel (12)]. Nevertheless, the diagrams of double row/column swaps, double multiplication of rows and n (α, β, f (γ)) satisfy the conditions of Theorem 4.4, and in fact columns by −1, and moves ±dij , where dij denotes the opera- define the standard genus 2 trisection of 2# 2. tion on αQγ which adds the ith column to the j th column, and CP CP then adds the ith row to the j th row, and −dij denotes the move ACKNOWLEDGMENTS. We thank Rob Kirby, Jeff Meier, and Matthias Nagel which subtracts rows and columns from each other. We leave it for helpful comments and conversations. We thank the American Institute of to the reader to verify that the first two moves can be realized by Mathematics and the organizers of its March 2017 conference on trisections, index changes and orientation changes, respectively, and in the without which this collaboration would not have occurred.

1. Rokhlin VA (1952) New results in the theory of 4-dimensional manifolds. Dokl Akad 7. Kosinski AA (1993) Differential Manifolds. Pure and Applied Mathematics (Academic, Nauk SSSR 4:221–224. New York), Vol 138. 2. Donaldson S (1983) An application of gauge theory to 4-dimensional topology. J Diff 8. Wall C (1969) Non-additivity of the signature. Inventiones Math 7:269–274. Geom 18:279–315. 9. Waldhausen F (1968) Heegaard-zerlegungen der 3-spaare.¨ Topology 7:195– 3. Freedman M (1982) The topology of 4-dimensional manifolds. J Diff Geom 17:357– 203. 453. 10. Meier J, Zupan A (2017) Bridge trisections of knotted surfaces in 4-manifolds. Trans 4. Gay D, Kirby R (2016) Trisecting 4-manifolds. Geom Topol 20:3097–3132. Am Math Soc 369:7343–7386. 5. Gompf R, Stipsicz A (1999) 4-Manifolds and Kirby Calculus. Graduate Studies in 11. Penner R (1988) A construction of pseudo-Anasov homeomorphisms. Trans Am Math Mathematics, (American Mathematical Society, Providence, RI). Vol 20. Soc 310:180–197. 6. Meier J, Schirmer T, Zupan A (2016) Classification of trisections and the generalized 12. Hempel J (2001) 3-manifolds as viewed form the curve complex. Topology 40:631– property R conjecture. Proc Am Math Soc 144:4983–4997. 657.

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