Computing trisections of 4-manifolds SPECIAL FEATURE Mark Bella, Joel Hassb,1, Joachim Hyam Rubinsteinc, and Stephan Tillmannd aDepartment of Mathematics, University of Illinois, Urbana, IL 61801; bDepartment of Mathematics, University of California, Davis, CA 95616; cSchool of Mathematics and Statistics, The University of Melbourne, Melbourne, VIC 3010, Australia; and dSchool of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, Australia Edited by David Gabai, Princeton University, Princeton, NJ, and approved July 9, 2018 (received for review October 23, 2017) We describe an algorithm to compute trisections of orientable trisection submanifolds. In our illustrations, we use the colors four-manifolds using arbitrary triangulations as input. This results blue, red, and green instead of zero, one, and two, and we will in explicit complexity bounds for the trisection genus of a 4- refer to Hblue red = Hbr as the green submanifold and so on. manifold in terms of the number of pentachora (4-simplices) in The above definition is somewhat more general than the one a triangulation. originally given by Gay and Kirby (14) in that they ask for the tri- section to be balanced in the sense that each handlebody Hi has four-manifold j trisection j four-dimensional triangulations j topological the same genus. It was noted in ref. 13 that any unbalanced tri- algorithms j computational topology section can be stabilized to a balanced one. We also remark that the above definition and our methods naturally extend to nonori- guiding principle in low-dimensional topology is to find entable 4-manifolds, but here, we restrict ourselves to the setting Apractical algorithms to describe topological or geometric of orientable manifolds following ref. 14. A representation of a structures on manifolds and to compute invariants as well as trisection, dropping down two dimensions, is shown in Fig. 1. to determine explicit complexity bounds for these algorithms. This representation completely encapsulates our approach. We The steps in an algorithm reveal the structure of a manifold, wish to define maps from 4-manifolds to the 2-simplex such that and the complexity bounds relate the relative difficulty of a the dual cubical structure of the 2-simplex pulls back to trisections wide variety of problems. This paper is a first step toward a of the 4-manifolds. computational theory for understanding 4-manifolds via trisec- We use singular triangulations to give a concrete description of a 4-manifold M . To induce a trisection of M , we use maps from tions. We use singular triangulations to give a description of a 2 MATHEMATICS 4-manifold. These are well-established as a data structure for M to the standard 2-simplex ∆ that are induced by what we algorithmic 3-manifold theory (1–9) and have shown promise call tricolorings. The aim of this note is to describe an algorithm for analyzing manifolds in higher dimensions (10–12). Budney to compute a trisection diagram on the central surface given an et al. (10) have developed a census of 4-manifold triangula- arbitrary singular triangulation of M and to obtain complexity tions with up to six pentachora. This is a rich source of exam- bounds on this description in terms of the size of the input trian- ples, and additional study or extension of this census requires gulation. The definitions are motivated by the example given in algorithmic tools. the next section and illustrated in Fig. 2. We develop a theory of colorings for 4-manifold triangula- tions, starting with a basic notion of tricoloring that encodes 2. Example suitable maps to the 2-simplex and enhancing this to c-tricoloring Consider the moment map from the complex projective plane 2 2 3 with appropriate connectivity properties and ts-tricoloring, which to the standard two-dimensional simplex µ: CP ! ∆ ⊂ R completely encodes a trisection. This refines previous work of defined by two of the authors (13). 1 1. Introduction [ z0 : z1 : z2 ] 7! P (jz0j, jz1j, jz2j): jzk j Gay and Kirby (14) introduced the concept of a trisection for arbi- trary smooth, oriented, closed 4-manifolds. In dimensions less The dual spine Π2 in ∆2 is the subcomplex of the first barycen- than or equal to four, there is a bijective correspondence between tric subdivision of ∆2 spanned by the 0-skeleton of the first isotopy classes of smooth and piecewise linear structures (15, 16). All manifolds are assumed to be piecewise linear (PL) and ori- entable in this paper unless stated otherwise. Our definition and Significance results apply to any compact smooth manifold by passing to its unique PL structure (17). Algorithms that decompose a manifold into simple pieces Definition 1 (trisection of closed manifold): Let M be a closed, reveal the geometric and topological structure of the man- orientable, connected PL 4-manifold. A trisection of M is a col- ifold, showing how complicated structures are constructed lection of three PL submanifolds H0, H1, H2 ⊂ M , subject to the from simple building blocks. This note describes a way to following four conditions. algorithmically construct a trisection, which describes a four- dimensional manifold as a union of three four-dimensional 1. Each Hi is PL homeomorphic to a standard PL four- 1-handlebodies. The complexity of the 4-manifold is captured dimensional 1–handlebody of genus gi . in a collection of curves on a surface, which guide the gluing of 2. The handlebodies Hi have pairwise disjoint interiors, and S the 1-handlebodies. The algorithm begins with a description M = i Hi . of a manifold as a union of pentachora or four-dimensional 3. The intersection Hi \ Hj of any two of the handlebodies is a simplices. It transforms this description into a trisection. three-dimensional 1–handlebody. 4. The common intersection Σ = H0 \ H1 \ H2 of all three Author contributions: M.B., J.H., J.H.R., and S.T. performed research and wrote the paper. handlebodies is a closed, connected surface, the central The authors declare no conflict of interest. surface. This article is a PNAS Direct Submission. The orientability of M implies that each of the handlebod- Published under the PNAS license. ies and the central surface in a trisection of M are orientable. 1 To whom correspondence should be addressed. Email: [email protected] The submanifolds Hij = Hi \ Hj and Σ are referred to as the www.pnas.org/cgi/doi/10.1073/pnas.1717173115 PNAS Latest Articles j 1 of 7 Downloaded by guest on September 30, 2021 Singular Triangulations. Let ∆e be a finite union of pairwise dis- joint, oriented Euclidean 4-simplices with the standard simplicial structure. We call a 4-simplex a pentachoron. Every k-simplex τ in ∆e is contained in a unique pentachoron στ . A 3-simplex in ∆e is called a facet, and a 0-simplex is a vertex. Let Φ be a family of orientation-reversing affine isomorphisms pairing the facets in ∆e , with the properties that ' 2 Φ if and only if '−1 2 Φ, and every facet is the domain of a unique element of Φ. The elements of Φ are called face pairings. We denote T = (∆,e Φ). Any operation O of simplicial topol- ogy that is performed on ∆e (such as barycentric subdivision, regular neighborhoods, and so on) is said to be an opera- tion on T as long as it respects the face pairings. The set of all face pairings Φ determines a natural equivalence rela- tion on the set of all k-simplices in ∆e , and we will call the equivalence classes the (singular) k-simplices of T . This termi- Fig. 1. Cartoon of a trisection. nology is natural when passing to the quotient space jT j = ∆e =Φ with the quotient topology. The space jT j is a closed, ori- entable four-dimensional pseudomanifold, and the quotient map barycentric subdivision minus the 0-skeleton of ∆2. Decom- is denoted p : ∆e ! jT j. The set of nonmanifold points of jT j, posing along Π2 gives ∆2 a natural dual cubical structure with if any, is contained in the 1-skeleton [the work by Seifert and three 2-cubes, and the lower-dimensional cubes that we will Threfall (18)]. focus on are the intersections of nonempty collections of these A singular triangulation of a 4-manifold M is a PL homeomor- top-dimensional cubes, consisting of three interior 1-cubes and phism jT j ! M , where jT j is obtained as above. The triangula- one interior 0-cube. The cubical structure is indicated in Fig. tion is simplicial if p : ∆e ! jT j is injective on each simplex. The 1, where the interior cubes are labeled with the trisection triangulation is PL if, in addition, the link of every simplex is PL submanifolds. homeomorphic to a standard sphere: @([0, 1]n ): Under the moment map, the 2-cubes pull back to 4-balls Tricolorings. Let M be a closed, connected 4-manifold with (pos- f[z0 : z1 : z2] j zi = 1, jzj j ≤ 1, jzk j ≤ 1g; sibly singular) triangulation jT j ! M . A partition fP0, P1, P2g T tricoloring 1 2 of the set of all vertices of is a if every pentachoron the interior 1-cubes pull back to solid tori S × D defined by meets two of the partition sets in two vertices and the remaining partition set in a single vertex. In this case, we also say that the f[z0 : z1 : z2] j zi = 1, jzj j = 1, jzk j ≤ 1g; triangulation is tricolored. 2 1 1 Denote the vertices of the standard 2-simplex ∆ by v0, v1, and the interior 0-cube pulls back to a 2-torus Σ = S × S 2 and v2.