[Complexité ombre et genre de trisection]
The shadow-complexity is an invariant of closed $4$-manifolds defined by using $2$-dimensional polyhedra called Turaev’s shadows, which, roughly speaking, measures how complicated a $2$-skeleton of the $4$-manifold is. In this paper, we define a new version $\mathrm{sc}_{r}$ of shadow-complexity depending on an extra parameter $r\ge 0$, and we investigate the relationship between this complexity and the trisection genus $g$. More explicitly, we prove an inequality $g(W) \le 2+2\mathrm{sc}_{r}(W)$ for any closed $4$-manifold $W$ and any $r\ge 1/2$. Moreover, we determine the exact values of $\mathrm{sc}_{1/2}$ for infinitely many $4$-manifolds, and also we classify all the closed $4$-manifolds with $\mathrm{sc}_{1/2}\le 1/2$.
La complexité ombre est un invariant de variétés de dimension $4$ défini en utilisant des polyèdres de dimension $2$, appelés les « ombres de Turaev », qui, de façon simplifiée, mesure la complexité d’un $2$-squelette de la $4$-variété. Dans cet article, nous définissons une version de la complexité ombre $\mathrm{sc}_{r}$ dépendant d’un paramètre supplémentaire $r\ge 0$, et nous investiguons les liens entre cette complexité et le genre de trisection $g$. Plus explicitement, nous prouvons l’inégalité $g(W) \le 2+2\mathrm{sc}_{r}(W)$ pour toute $4$-variété fermée et tout $r\ge 1/2$. De plus, nous déterminons les valeurs exactes de $\mathrm{sc}_{1/2}$ pour une famille infinie de $4$-variétés, et nous classifions toutes les $4$-variétés fermées avec $\mathrm{sc}_{1/2}\le 1/2$.
Accepté le :
Publié le :
Keywords: $4$-manifolds, shadows, shadow-complexity, trisections, trisection genus
Hironobu Naoe 1 ; Masaki Ogawa 2
CC-BY 4.0
@article{AFST_2025_6_34_5_1185_0,
author = {Hironobu Naoe and Masaki Ogawa},
title = {Shadow-complexity and trisection genus},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {1185--1217},
year = {2025},
publisher = {Universit\'e de Toulouse, Toulouse},
volume = {Ser. 6, 34},
number = {5},
doi = {10.5802/afst.1830},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1830/}
}
TY - JOUR AU - Hironobu Naoe AU - Masaki Ogawa TI - Shadow-complexity and trisection genus JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2025 SP - 1185 EP - 1217 VL - 34 IS - 5 PB - Université de Toulouse, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1830/ DO - 10.5802/afst.1830 LA - en ID - AFST_2025_6_34_5_1185_0 ER -
%0 Journal Article %A Hironobu Naoe %A Masaki Ogawa %T Shadow-complexity and trisection genus %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2025 %P 1185-1217 %V 34 %N 5 %I Université de Toulouse, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1830/ %R 10.5802/afst.1830 %G en %F AFST_2025_6_34_5_1185_0
Hironobu Naoe; Masaki Ogawa. Shadow-complexity and trisection genus. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 5, pp. 1185-1217. doi: 10.5802/afst.1830
[1] Shadows and branched shadows of and -manifolds, Tesi. Scuola Normale Superiore di Pisa (Nuova Serie), 1, Edizioni della Normale, 2005 | Zbl
[2] Complexity of 4-manifolds, Exp. Math., Volume 15 (2006) no. 2, pp. 237-249 | DOI | Zbl | MR
[3] Stein domains and branched shadows of 4-manifolds, Geom. Dedicata, Volume 121 (2006), pp. 89-111 | DOI | Zbl | MR
[4] Branched shadows and complex structures on 4-manifolds, J. Knot Theory Ramifications, Volume 17 (2008) no. 11, pp. 1429-1454 | DOI | Zbl | MR
[5] 3-manifolds efficiently bound 4-manifolds, J. Topol., Volume 1 (2008) no. 3, pp. 703-745 | DOI | Zbl | MR
[6] Trisecting 4-manifolds, Geom. Topol., Volume 20 (2016) no. 6, pp. 3097-3132 | DOI | Zbl | MR
[7] 4-manifolds and Kirby calculus, Graduate Studies in Mathematics, 20, American Mathematical Society, 1999 | Zbl | MR
[8] Stable maps and branched shadows of 3-manifolds, Math. Ann., Volume 367 (2017) no. 3-4, pp. 1819-1863 | DOI | Zbl | MR
[9] Four-manifolds with shadow-complexity one, Ann. Fac. Sci. Toulouse, Math. (6), Volume 31 (2022) no. 4, pp. 1111-1212 | Zbl | Numdam | MR | DOI
[10] Shadows of acyclic 4-manifolds with sphere boundary, Algebr. Geom. Topol., Volume 20 (2020) no. 7, pp. 3707-3731 | DOI | Zbl | MR
[11] A note on 4-dimensional handlebodies, Bull. Soc. Math. Fr., Volume 100 (1972), pp. 337-344 | DOI | Zbl | Numdam | MR
[12] Links, two-handles, and four-manifolds, Int. Math. Res. Not., Volume 2005 (2005) no. 58, pp. 3595-3623 | DOI | Zbl | MR
[13] Four-manifolds with shadow-complexity zero, Int. Math. Res. Not., Volume 2011 (2011) no. 6, pp. 1268-1351 | DOI | Zbl | MR
[14] Trisections and spun four-manifolds, Math. Res. Lett., Volume 25 (2018) no. 5, pp. 1497-1524 | DOI | Zbl | MR
[15] Classification of trisections and the Generalized Property R Conjecture, Proc. Am. Math. Soc., Volume 144 (2016) no. 11, pp. 4983-4997 | DOI | Zbl | MR
[16] Genus-two trisections are standard, Geom. Topol., Volume 21 (2017) no. 3, pp. 1583-1630 | Zbl | DOI | MR
[17] Bridge trisections of knotted surfaces in 4-manifolds, Proc. Natl. Acad. Sci. USA, Volume 115 (2018) no. 43, pp. 10880-10886 | DOI | Zbl | MR
[18] Shadows of 4-manifolds with complexity zero and polyhedral collapsing, Proc. Am. Math. Soc., Volume 145 (2017) no. 10, pp. 4561-4572 | DOI | Zbl | MR
[19] The special shadow-complexity of , J. Knot Theory Ramifications, Volume 34 (2025) no. 5, 2550014, 12 pages | DOI | MR | Zbl
[20] Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics, 18, Walter de Gruyter, 2016 | DOI | Zbl | MR
[21] Trisections of Flat Surface Bundles over Surfaces, Ph. D. Thesis, University of Nebraska, Lincoln, USA (2020) (https://digitalcommons.unl.edu/dissertations/AAI28086334)
[22] Dehn surgery on arborescent links, Trans. Am. Math. Soc., Volume 351 (1999) no. 6, pp. 2275-2294 | DOI | Zbl | MR
Cité par Sources :