Genus-two trisections are standard

Meier, Jeffrey; Zupan, Alexander

doi:10.2140/gt.2017.21.1583

Cited by 33 publications

(34 citation statements)

References 15 publications

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“…Manifolds with trisection genus at most one are easy to classify [10]. In [24], it was shown that S 2 × S 2 is the unique irreducible 1 manifold with trisection genus two, and it was asked to what extent it is possible to enumerate manifolds with trisection genus g for low values of g. To this end, we offer the following conjecture. Conjecture 1.1.…”

Section: Outlinementioning

confidence: 99%

Trisections and spun four-manifolds

Meier

2018

Mathematical Research Letters

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We study trisections of 4-manifolds obtained by spinning and twist-spinning 3-manifolds, and we show that, given a (suitable) Heegaard diagram for the 3-manifold, one can perform simple local modifications to obtain a trisection diagram for the 4-manifold. We also show that this local modification can be used to convert a (suitable) doubly-pointed Heegaard diagram for a 3-manifold/knot pair into a doubly-pointed trisection diagram for the 4-manifold/2-knot pair resulting from the twist-spinning operation.This technique offers a rich list of new manifolds that admit trisection diagrams that are amenable to study. We formulate a conjecture about 4-manifolds with trisection genus three and provide some supporting evidence. 1 We call a 4-manifold X irreducible if each summand of any connected sum decomposition of X is either X or a homotopy 4-sphere.

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Section: Outlinementioning

confidence: 99%

Trisections and spun four-manifolds

Meier

2018

Mathematical Research Letters

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“…If K admits a 3-bridge trisection, then Σ 2 (K) admits a 2-trisection, as discussed above in Subsection 2.6. In [24], it is shown that every balanced 2-trisection is standard, and in [23] the unbalanced case is resolved. These results imply a classification of 3-bridge surfaces.…”

Section: Classifying 3-bridge Trisectionsmentioning

confidence: 99%

“…Gay and Kirby prove that every X admits a trisection, and any two trisections of X are related by natural stabilization and destabilization operations in a 4-dimensional version of the Reidemeister-Singer Theorem for Heegaard splittings of 3-manifolds. As evidence of the utility of trisections Date: July 31, 2015. to bridge the gap between 3-and 4-manifolds, the authors have shown in previous work [24] that 3-dimensional tools suffice to classify those 4-manifolds which admit a trisection of genus two.…”

Section: Introductionmentioning

confidence: 99%

“…Given a knotted surface K in S 4 , we can consider the double cover X(K) of S 4 branched over K. A (b; c 1 , c 2 , c 3 )-bridge trisection of K gives rise to a (b − 1; c 1 − 1, c 2 − 1, c 3 − 1)-trisection of the closed 4-manifold X(K). Theorems in [23] and [24] classify balanced and unbalanced genus two trisections of 4-manifolds, respectively. In particular, every genus two trisection is standard, and we obtain the following result as a corollary.…”

Section: Introductionmentioning

confidence: 99%

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Bridge trisections of knotted surfaces in 𝑆⁴

Meier

Zupan

2017

Trans. Amer. Math. Soc.

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We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of fourmanifolds and extends the classical concept of bridge splittings of links in the threesphere to four dimensions. We prove that every knotted surface in the four-sphere admits a bridge trisection (a decomposition into three simple pieces) and that any two bridge trisections for a fixed surface are related by a sequence of stabilizations and destabilizations. We also introduce a corresponding diagrammatic representation of knotted surfaces and describe a set of moves that suffice to pass between two diagrams for the same surface. Using these decompositions, we define a new complexity measure: the bridge number of a knotted surface. In addition, we classify bridge trisections with low complexity, we relate bridge trisections to the fundamental groups of knotted surface complements, and we prove that there exist knotted surfaces with arbitrarily large bridge number. arXiv:1507.08370v1 [math.GT]

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“…Figure 2 illustrates the same diagram more topologically. (For trisections with g = 1 and g = 2, see the basic 4-manifold trisection examples in [1]; in fact, the 4dimensional uniqueness results in [4], together with Theorem 5 below, give uniqueness statements for group trisections with g ≤ 2.) Definition 2 Given a (g, k)-trisection ({G v }, {f e }) of G and a (g , k )-trisection ({G v }, {f e }) of G , there is a natural "connected sum" (g = g + g , k = k + k )trisection ({G v }, {f e }) of G = G * G defined by first shifting all the indices of the generators for the G v 's by either g (when G v = S g or G v = H g ) or k (when G v = Z k ) and then, for each generator y of G v , declaring f e (y) to be either f e (y) or f e (y) according to whether y is in G v or G v .…”

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confidence: 99%

Group trisections and smooth 4–manifolds

Abrams

Kirby

2018

Geom. Topol.

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A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the closed 4-manifold, with homomorphisms between them induced by inclusion, form a commutative diagram of epimorphisms, which we call a trisection of the 4-manifold group. A trisected 4-manifold thus gives a trisected group; here we show that every trisected group uniquely determines a trisected 4-manifold. Together with Gay and Kirby's existence and uniqueness theorem for 4-manifold trisections, this gives a bijection from group trisections modulo isomorphism and a certain stabilization operation to smooth, closed, connected, oriented 4-manifolds modulo diffeomorphism. As a consequence, smooth 4-manifold topology is, in principle, entirely group theoretic. For example, the smooth 4-dimensional Poincaré conjecture can be reformulated as a purely group theoretic statement. * 57M05; 20F05 Let g and k be integers with g ≥ k ≥ 0. We fix the following groups, described explicitly by presentations:standard genus g surface group with standard labelled generators. We identify this in the obvious way with π 1 (# g S 1 × S 1 , * ).• H 0 = {1} and, for g > 0, H g = x 1 , . . . , x g , i.e. a free group of rank g with g labelled generators. We identify this in the obvious way with π 1 ( g S 1 × B 2 , * ). Note that, if g < g , then H g ⊂ H g .

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Genus-two trisections are standard

Cited by 33 publications

References 15 publications

Trisections and spun four-manifolds

Trisections and spun four-manifolds

Bridge trisections of knotted surfaces in 𝑆⁴

Group trisections and smooth 4–manifolds

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