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Calculating the Homology and Intersection Form of a 4-Manifold from a Trisection Diagram

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Calculating the Homology and Intersection Form of a 4-Manifold from a Trisection Diagram Calculating the homology and intersection form of a SPECIAL FEATURE 4-manifold from a trisection diagram Peter Fellera,1, Michael Klugb,1, Trenton Schirmerc,1,2, and Drew Zemked,1 aDepartment of Mathematics, ETH Zurich, 8092 Zurich, Switzerland; bDepartment of Mathematics, University of California, Berkeley, CA 94720; cDepartment of Mathematics, University of Georgia, Athens, GA 30602; and dDepartment of Mathematics, Cornell University, Ithaca, NY 14850 Edited by David Gabai, Princeton University, Princeton, NJ, and approved June 15, 2018 (received for review October 31, 2017) Given a diagram for a trisection of a 4-manifold X, we describe from this new perspective. More generally, we expect the calcu- the homology and the intersection form of X in terms of the three lation of these elementary yet fundamental invariants to play an subgroups of H1(F; Z) generated by the three sets of curves and important role in the continued development of the theory of the intersection pairing on H1(F; Z). This includes explicit formulas trisections. for the second and third homology groups of X as well an algo- rithm to compute the intersection form. Moreover, we show that Results. Let (Σ; V1, V2, V3) denote a (g; k1, k2, k3)-trisection all (g; k, 0, 0)-trisections admit “algebraically trivial” diagrams. of a smooth closed oriented 4-manifold X , where Vi is the ki 1 3 genus ki (4, 1)-handlebody \ S × D and Σ = V1 \ V2 \ V3 4-manifolds j trisections j homology j intersection form is a closed orientable genus g surface. Let Wα = V1 \ V2, Wβ = V1 \ V3, and Wγ = V2 \ V3, each of which is a genus g 1 2 g (3, 1)-handlebody \ S × D . For 2 fα, β, γg let L = ker(ι), n this note, we describe how to compute the homology and where ι : H1(Σ) ! H1(W) is the map induced by inclusion. Iintersection form of a 4-manifold using the algebraic intersec- The groups L are maximal Lagrangian subgroups of H1(Σ) tion numbers of the curves appearing in any trisection diagram generated by any choice of oriented defining curves for W. for the 4-manifold. Moreover, we show that a large class of We show that the following chain complex gives rise to the trisections admit “algebraically trivial” diagrams. homology of X . (The groups depicted are in dimensions four through zero, and the complex is understood to have trivial MATHEMATICS 1. Introduction groups and maps in other dimensions.) Homology and the Intersection Form. The homology of an n- dimensional manifold X is a collection of n + 1 abelian groups 0 @3 Lγ @2 0 H0(X ), H1(X ), ::: , Hn (X ). One may think of the elements of Z −! Lα \ Lγ −! −! Hom(Lα \ Lβ , Z) −! Z: [1.1] Lβ \ Lγ Hi (X ) as equivalence classes of oriented closed i-dimensional submanifolds immersed in X —where two such i-manifolds The map @3 is induced by the inclusion Lα \ Lγ ,! Lγ and @2 is Y1 and Y2 are considered equivalent if there is an (i + 1)- given by [x] 7! h·, xiΣ (h·, ·iΣ denotes the intersection form on dimensional manifold immersed in X which is bounded by Y1 H1(Σ, Z)). and Y2. If Σ is a closed surface, for example, then the ele- We can also describe the homology of X in a way that is ments of H1(Σ) can be thought of as immersed oriented curves symmetric in the three Lagrangian subspaces L. As shown in in Σ, where an orientation in this case amounts to assign- Corollary 3.3, the second homology group of X can be identified ing a direction to a curve. Two sets of curves cancel one with the group another out if they bound an immersed subsurface with orien- tations which match the boundary orientation of the surface. fa + b + c = 0g Addition of homology classes corresponds to taking the union , of curves. fc = 0g + fb = 0g + fa = 0g In the case of a 4-manifold X , the elements of H2(X ) cor- respond to embedded surfaces inside of X . Surfaces embedded where fa + b + c = 0g denotes the subgroup of Lα × Lβ × Lγ of in a 4-manifold generically intersect one another in points. (To elements (a, b, c) such that a + b + c = 0, and the summands in see why, consider the intersection of the x − y plane with the 4 the denominators are likewise understood to be the subgroups z − w plane in R ; in fact, this is an appropriate local model for a of fa + b + c = 0g satisfying the given linear equations. With generic intersection of surfaces in a 4-manifold.) The intersection form of a 4-manifold is a symmetric, bilinear product on H2(X ) which, given a pair of surfaces Σ1 and Σ2 as input, outputs the Significance “algebraic count” of the intersections between them. When trying to prove that two topological spaces are distinct, Motivation. Whenever a new description of a topological object one often associates algebraic invariants such as groups, rings, arises, it is natural to ask how this can be used to describe the fun- and matrices to topological spaces. For manifolds, in dimen- damental algebraic invariants of the space, such as its homology sion four, the homology and intersection form are especially and intersection form. The homology and intersection forms of useful invariants. Here, we describe how they can be com- 4-manifolds reveal an interesting dichotomy between the smooth puted using the information contained in a trisection diagram and the topological category of 4-manifolds. On the one hand, and provide some applications. results such as Rokhlin’s theorem (1) and Donaldson’s diagonal- ization theorem (2) impose strong restrictions on the intersection Author contributions: P.F., M.K., T.S., and D.Z. designed research; P.F., M.K., T.S., and D.Z. form of smooth closed oriented 4-manifolds; but on the other performed research; and P.F., T.S., and D.Z. wrote the paper. hand, all symmetric unimodular bilinear forms occur as the inter- The authors declare no conflict of interest. section form of some topological closed oriented 4-manifolds, by Freedman’s work (3). Since every smooth 4-manifold admits a This article is a PNAS Direct Submission. trisection (4), it is worthwhile to be able to compute the homol- Published under the PNAS license. ogy and the intersection form of a 4-manifold directly from a 1 P.F., M.K., T.S., and D.Z. contributed equally to this work.y trisection diagram. We hope that Donaldson’s strong restrictions 2 To whom correspondence should be addressed. Email: [email protected] on intersection forms will also eventually become more apparent www.pnas.org/cgi/doi/10.1073/pnas.1717176115 PNAS Latest Articles j 1 of 6 Downloaded by guest on October 1, 2021 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 fα1, ::: , αg g is any defining set of curves for W in Σ, then 0 1 0 −1 0 1 f[α1], ::: ,[αg ]g forms a basis of ker(ι∗), and αQβ = −1 0 , γ Qβ = −1 0 , αQγ = −1 0 : 2. For all x, y 2 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 fα1, ::: , αg g is a defining set of curves, the set f[α1], ::: ,[αg ]g 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 2 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.
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