Bridge trisections in rational surfaces

Lambert-Cole, Peter; Meier, Jeffrey

doi:10.1142/s1793525321500047

Cited by 11 publications

(32 citation statements)

References 19 publications

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“…Gay and Kirby showed that every closed, oriented, smooth 4‐manifold

X

admits a trisection

double-struckT

and that any two trisections

double-struckT

and

T^{'}

X

have a common stabilization [12]. Trisections of low genus have been classified [22, 23], and trisections for many familiar 4‐manifolds, including complex hypersurfaces in

{CP}^{3}

, have been described [21].…”

Section: Definitions Notions Of Equivalence and Open Questionsmentioning

confidence: 99%

“…Example Consider

{CP}^{2}

with homogeneous coordinates

[z_{1} : z_{2} : z_{3}]

, and consider the subsets

Z_{λ} = false{false[z_{1} : z_{2} : z_{3} false] ∣ false| z_{λ} false|, false| z_{λ + 1} false| ⩽ false| z_{λ + 2} false| false}

H_{λ} = false{false[z_{1} : z_{2} : z_{3} false] ∣ false| z_{λ} false| ⩽ false| z_{λ + 1} false| = false| z_{λ + 2} false| false} .

This decomposition yields a Weinstein trisection

T_{0}

for

({double-struckCP}^{2}, ω_{F S})

first given in [21], where

ω_{F S}

denotes the Fubini–Study symplectic form. This trisection is compatible with the toric structure on

{CP}^{2}

, and Figure 1 shows the decomposition in the image of the moment map.…”

Section: Definitions Notions Of Equivalence and Open Questionsmentioning

confidence: 99%

“…Note that there are examples of symplectic 4‐manifolds whose Weinstein trisection genus (for some symplectic form) equals their trisection genus. These include complex hypersurfaces in

{CP}^{3}

and the elliptic surfaces

E (n)

; compare [21, Theorem 1.1] with Theorem 6.1 and Example 7.1.…”

Section: Definitions Notions Of Equivalence and Open Questionsmentioning

confidence: 99%

“…We call

scriptD

a singular disk‐tangle for

L

if (1)each component of

scriptD

with unknotted boundary is a smooth, properly embedded, boundary parallel disk; (2)for each component

D

scriptD

with knotted boundary

K

, there is a 4‐ball

B \subset Z

such that

B \cap S^{3}

is a 3‐ball containing

K

and

D

is a cone

K

B

and (3)the components of

scriptD

are disjoint when their boundaries are split and intersect transversely — away from all cone points — otherwise. For example, if

L

were the split union of a the torus link

T (9, 6)

, whose components are three trefoils, and the Hopf link

T (2, 1)

, then

scriptD

would consist of three cones on trefoils, which intersect pairwise transversely in six points, split union two smoothly embedded disks, which intersect transversely in a single point. Note that this is a generalization of the notion of a trivial disk‐tangle appearing elsewhere in the literature [6, 21, 24, 25]. Definition Let

X

be a 4‐manifold equipped with a trisection

double-struckT

, and let

S \subset X

be a singular surface.…”

Section: Singular Bridge Trisections and Branched Coveringsmentioning

confidence: 99%

“…The proof of Theorem 1.1 relies on important results of Auroux [2] and Auroux–Katzarkov [3] that every closed, symplectic 4‐manifold admits a quasiholomorphic branched covering map

f : X \to {double-struckCP}^{2}

. Trisections are naturally suited for studying branched coverings of 4‐manifolds (for example, [5, 6, 21, 27]), and the basic strategy is to pull back the Weinstein trisection of

{CP}^{2}

via the singular branched covering map

f

.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Symplectic 4‐manifolds admit Weinstein trisections

Lambert-Cole

Meier

Starkston

2021

Journal of Topology

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We prove that every symplectic 4‐manifold admits a trisection that is compatible with the symplectic structure in the sense that the symplectic form induces a Weinstein structure on each of the three sectors of the trisection. Along the way, we show that a (potentially singular) symplectic braided surface in CP2 can be symplectically isotoped into bridge position. This paper relies extensively on colour figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the colour figures.

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“…Gay and Kirby showed that every closed, oriented, smooth 4‐manifold

X

admits a trisection

double-struckT

and that any two trisections

double-struckT

and

T^{'}

X

have a common stabilization [12]. Trisections of low genus have been classified [22, 23], and trisections for many familiar 4‐manifolds, including complex hypersurfaces in

{CP}^{3}

, have been described [21].…”

Section: Definitions Notions Of Equivalence and Open Questionsmentioning

confidence: 99%

“…Example Consider

{CP}^{2}

with homogeneous coordinates

[z_{1} : z_{2} : z_{3}]

, and consider the subsets

Z_{λ} = false{false[z_{1} : z_{2} : z_{3} false] ∣ false| z_{λ} false|, false| z_{λ + 1} false| ⩽ false| z_{λ + 2} false| false}

H_{λ} = false{false[z_{1} : z_{2} : z_{3} false] ∣ false| z_{λ} false| ⩽ false| z_{λ + 1} false| = false| z_{λ + 2} false| false} .

This decomposition yields a Weinstein trisection

T_{0}

for

({double-struckCP}^{2}, ω_{F S})

first given in [21], where

ω_{F S}

denotes the Fubini–Study symplectic form. This trisection is compatible with the toric structure on

{CP}^{2}

, and Figure 1 shows the decomposition in the image of the moment map.…”

Section: Definitions Notions Of Equivalence and Open Questionsmentioning

confidence: 99%

“…Note that there are examples of symplectic 4‐manifolds whose Weinstein trisection genus (for some symplectic form) equals their trisection genus. These include complex hypersurfaces in

{CP}^{3}

and the elliptic surfaces

E (n)

; compare [21, Theorem 1.1] with Theorem 6.1 and Example 7.1.…”

Section: Definitions Notions Of Equivalence and Open Questionsmentioning

confidence: 99%

“…We call

scriptD

a singular disk‐tangle for

L

if (1)each component of

scriptD

with unknotted boundary is a smooth, properly embedded, boundary parallel disk; (2)for each component

D

scriptD

with knotted boundary

K

, there is a 4‐ball

B \subset Z

such that

B \cap S^{3}

is a 3‐ball containing

K

and

D

is a cone

K

B

and (3)the components of

scriptD

are disjoint when their boundaries are split and intersect transversely — away from all cone points — otherwise. For example, if

L

were the split union of a the torus link

T (9, 6)

, whose components are three trefoils, and the Hopf link

T (2, 1)

, then

scriptD

X

be a 4‐manifold equipped with a trisection

double-struckT

, and let

S \subset X

be a singular surface.…”

Section: Singular Bridge Trisections and Branched Coveringsmentioning

confidence: 99%

“…The proof of Theorem 1.1 relies on important results of Auroux [2] and Auroux–Katzarkov [3] that every closed, symplectic 4‐manifold admits a quasiholomorphic branched covering map

f : X \to {double-struckCP}^{2}

. Trisections are naturally suited for studying branched coverings of 4‐manifolds (for example, [5, 6, 21, 27]), and the basic strategy is to pull back the Weinstein trisection of

{CP}^{2}

via the singular branched covering map

f

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Symplectic 4‐manifolds admit Weinstein trisections

Lambert-Cole

Meier

Starkston

2021

Journal of Topology

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Symplectic surfaces and bridge position

Lambert-Cole¹

2022

Geom Dedicata

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Simplifying indefinite fibrations on 4-manifolds

Baykur

Saeki

2023

Trans. Amer. Math. Soc.

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The main goal of this article is to connect some recent perspectives in the study of 4 4 -manifolds from the vantage point of singularity theory. We present explicit algorithms for simplifying the topology of various maps on 4 4 -manifolds, which include broken Lefschetz fibrations and indefinite Morse 2 2 -functions. The algorithms consist of sequences of moves, which modify indefinite fibrations in smooth 1 1 -parameter families. These algorithms allow us to give purely topological and constructive proofs of the existence of simplified broken Lefschetz fibrations and Morse 2 2 -functions on general 4 4 -manifolds, and a theorem of Auroux–Donaldson–Katzarkov on the existence of certain broken Lefschetz pencils on near-symplectic 4 4 -manifolds. We moreover establish a correspondence between broken Lefschetz fibrations and Gay–Kirby trisections of 4 4 -manifolds, and show the existence and stable uniqueness of simplified trisections on all 4 4 -manifolds. Building on this correspondence, we also provide several new constructions of trisections, including infinite families of genus- 3 3 trisections with homotopy inequivalent total spaces, and exotic same genera trisections of 4 4 -manifolds in the homeomorphism classes of complex rational surfaces.

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Bridge trisections in rational surfaces

Cited by 11 publications

References 19 publications

Symplectic 4‐manifolds admit Weinstein trisections

Symplectic 4‐manifolds admit Weinstein trisections

Symplectic surfaces and bridge position

Simplifying indefinite fibrations on 4-manifolds

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