Smoothly knotted and topologically unknotted nullhomologous surfaces in 4-manifolds

Torres, Rafael

doi:10.5802/aif.3685

Smoothly knotted and topologically unknotted nullhomologous surfaces in 4-manifolds
[Surfaces homologiquement triviales lissement nouées et topologiquement dénouées dans des variétés de dimension 4]

Torres, Rafael ¹

¹ SISSA Via Bonomea 265 34136 Trieste (Italy)

Annales de l'Institut Fourier, Online first, 27 p.

Résumé
Abstract

On remarque que des constructions récentes de structures lisses non équivalentes donnent lieu à une procédure de fabrication d’ensembles infinis de 2-sphères et de 2-tores homologiquement triviaux, topologiquement isotopes et deux à deux lissement non équivalent, à l’intérieur d’une myriade de 4-variétés. Ces surfaces bordent un corps d’anses plongé localement plat, maîs qui ne peut pas être plongé de manière lisse, et incluent les premiers exemples de 2-sphères homotopiquement triviales dans des 4-variétés fermées ayant un tel comportement. Nous donnons un exemple d’un plongement localement plat non lissifiable d’une 2-sphère homotopiquement triviale dans une 4-variété fermée, lisse et simplement connexe.

We point out that recent constructions of inequivalent smooth structures yield a manufacturing procedure of infinite sets of topologically isotopic and pairwise smoothly inequivalent nullhomologous 2-spheres and 2-tori inside a myriad of 4-manifolds. These surfaces bound a locally flat embedded handlebody but not a smoothly embedded one, and include the first examples of nullhomotopic 2-spheres in closed 4-manifolds to display such behavior. We exhibit an example of a locally flat non-smoothable embedding of a nullhomotopic 2-sphere in a closed smooth simply connected 4-manifold.

Reçu le : 2022-04-09
Révisé le : 2023-03-31
Accepté le : 2023-09-13
Première publication : 2025-02-10

DOI : 10.5802/aif.3685

Classification : 57K45, 57R55, 57R40, 57R52
Keywords: Surfaces, 4-manifolds, smooth structures
Mots-clés : Surfaces, variétés de dimension 4, structures lisses

Affiliations des auteurs :

Torres, Rafael ¹

¹ SISSA Via Bonomea 265 34136 Trieste (Italy)

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     title = {Smoothly knotted and topologically unknotted nullhomologous surfaces in 4-manifolds},
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Torres, Rafael. Smoothly knotted and topologically unknotted nullhomologous surfaces in 4-manifolds. Annales de l'Institut Fourier, Online first, 27 p.

Bibliographie
Cité par

[1] Akbulut, Selman The Catanese–Ciliberto–Mendes Lopes surface, J. Gökova Geom. Topol. GGT, Volume 5 (2011), pp. 86-102 | MR | Zbl

[2] Akbulut, Selman The Akhmedov–Park exotic ${ℂℙ}^{2} # 2 {\bar{ℂℙ}}^{2}$ , Adv. Math., Volume 274 (2015), pp. 948-967 | DOI | MR | Zbl

[3] Akbulut, Selman The Akhmedov–Park exotic ${ℂℙ}^{2} # 3 {\bar{ℂℙ}}^{2}$ , Adv. Math., Volume 274 (2015), pp. 928-947 | DOI | MR | Zbl

[4] Akbulut, Selman Isotoping 2-spheres in 4-manifolds, Proceedings of the Gökova Geometry-Topology Conference 2014, International Press (2015), pp. 264-266 | MR | Zbl

[5] Akbulut, Selman 4-manifolds, Oxf. Grad. Texts Math., 25, Oxford University Press, 2016, xii+262 pages | DOI | MR | Zbl

[6] Akhmedov, Anar; Baykur, R. İnanç; Park, B. Doug Constructing infinitely many smooth structures on small 4-manifolds, J. Topol., Volume 1 (2008) no. 2, pp. 409-428 | DOI | MR | Zbl

[7] Akhmedov, Anar; Park, B. Doug Exotic smooth structures on small 4-manifolds, Invent. Math., Volume 173 (2008) no. 1, pp. 209-223 | DOI | MR | Zbl

[8] Akhmedov, Anar; Park, B. Doug Exotic smooth structures on small 4-manifolds with odd signatures, Invent. Math., Volume 181 (2010) no. 3, pp. 577-603 | DOI | MR | Zbl

[9] Auckly, Dave; Kim, Hee Jung; Melvin, Paul; Ruberman, Daniel Stable isotopy in four dimensions, J. Lond. Math. Soc., Volume 91 (2015) no. 2, pp. 439-463 | DOI | MR | Zbl

[10] Auroux, D.; Donaldson, S. K.; Katzarkov, L. Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves, Math. Ann., Volume 326 (2003) no. 1, pp. 185-203 | DOI | MR | Zbl

[11] Baldridge, Scott; Kirk, Paul A symplectic manifold homeomorphic but not diffeomorphic to ${ℂℙ}^{2} # 3 \bar{{ℂℙ}^{2}}$ , Geom. Topol., Volume 12 (2008) no. 2, pp. 919-940 | DOI | MR | Zbl

[12] Baldridge, Scott; Kirk, Paul Constructions of small symplectic 4-manifolds using Luttinger surgery, J. Differ. Geom., Volume 82 (2009) no. 2, pp. 317-361 | DOI | MR | Zbl

[13] Baykur, R. İnanç; Sunukjian, Nathan Round handles, logarithmic transforms and smooth 4-manifolds, J. Topol., Volume 6 (2013) no. 1, pp. 49-63 | DOI | MR | Zbl

[14] Cochran, Tim D.; Habegger, Nathan On the homotopy theory of simply connected four manifolds, Topology, Volume 29 (1990) no. 4, pp. 419-440 | DOI | MR | Zbl

[15] Conway, Anthony; Powell, Mark Embedded surfaces with infinite cyclic knot group, Geom. Topol., Volume 27 (2023) no. 2, pp. 739-821 | DOI | MR | Zbl

[16] Finashin, Sergey Knotting of algebraic curves in $m a t h b b C P^{2}$ , Topology, Volume 41 (2002) no. 1, pp. 47-55 | DOI | MR | Zbl

[17] Fintushel, Ronald; Park, B. Doug; Stern, Ronald J. Reverse engineering small 4-manifolds, Algebr. Geom. Topol., Volume 7 (2007), pp. 2103-2116 | DOI | MR | Zbl

[18] Fintushel, Ronald; Stern, Ronald J. A fake $4$ -manifold with $π_{1} = Z$ and $b^{+} = 4$ , Turk. J. Math., Volume 18 (1994) no. 1, pp. 1-6 | MR | Zbl

[19] Fintushel, Ronald; Stern, Ronald J. Surfaces in $4$ -manifolds, Math. Res. Lett., Volume 4 (1997) no. 6, pp. 907-914 | DOI | MR | Zbl

[20] Fintushel, Ronald; Stern, Ronald J. Knots, links, and $4$ -manifolds, Invent. Math., Volume 134 (1998) no. 2, pp. 363-400 | DOI | MR | Zbl

[21] Fintushel, Ronald; Stern, Ronald J. Invariants for Lagrangian tori, Geom. Topol., Volume 8 (2004), pp. 947-968 | DOI | MR | Zbl

[22] Fintushel, Ronald; Stern, Ronald J. Pinwheels and nullhomologous surgery on 4-manifolds with $b^{+} = 1$ , Algebr. Geom. Topol., Volume 11 (2011) no. 3, pp. 1649-1699 | DOI | MR | Zbl

[23] Fintushel, Ronald; Stern, Ronald J. Surgery on nullhomologous tori, Proceedings of the Freedman Fest (Geometry and Topology Monographs), Volume 18, Geometry and Topology Publications (2012), pp. 61-81 | DOI | MR | Zbl

[24] Freedman, Michael H.; Quinn, Frank Topology of 4-manifolds, Princeton Mathematical Series, 39, Princeton University Press, 1990, viii+259 pages | MR | Zbl

[25] Friedl, Stefan; Hambleton, Ian; Melvin, Paul; Teichner, Peter Non-smoothable four-manifolds with infinite cyclic fundamental group, Int. Math. Res. Not., Volume 2007 (2007) no. 11, rnm031, 20 pages | DOI | MR | Zbl

[26] Gompf, Robert E. Sums of elliptic surfaces, J. Differ. Geom., Volume 34 (1991) no. 1, pp. 93-114 | DOI | MR | Zbl

[27] Gompf, Robert E. A new construction of symplectic manifolds, Ann. Math., Volume 142 (1995) no. 3, pp. 527-595 | DOI | MR | Zbl

[28] Gompf, Robert E.; Stipsicz, András I. $4$ -manifolds and Kirby calculus, Graduate Studies in Mathematics, 20, American Mathematical Society, 1999, xvi+558 pages | DOI | MR | Zbl

[29] Gordon, C. McA Knots in the $4$ -sphere, Comment. Math. Helv., Volume 51 (1976) no. 4, pp. 585-596 | DOI | MR | Zbl

[30] Hambleton, Ian; Teichner, Peter A non-extended Hermitian form over $ℤ [ℤ]$ , Manuscr. Math., Volume 93 (1997) no. 4, pp. 435-442 | DOI | MR | Zbl

[31] Hayden, Kyle Exotically knotted disks and complex curves (2020) | arXiv

[32] Ho, Chung-I; Li, Tian-Jun Luttinger surgery and Kodaira dimension, Asian J. Math., Volume 16 (2012) no. 2, pp. 299-318 | DOI | MR | Zbl

[33] Hoffman, Neil R.; Sunukjian, Nathan S. Null-homologous exotic surfaces in 4-manifolds, Algebr. Geom. Topol., Volume 20 (2020) no. 5, pp. 2677-2685 | DOI | MR | Zbl

[34] Juhász, András; Miller, Maggie; Zemke, Ian Transverse invariants and exotic surfaces in the 4-ball, Geom. Topol., Volume 25 (2021) no. 6, pp. 2963-3012 | DOI | MR | Zbl

[35] Kim, Hee Jung Modifying surfaces in 4-manifolds by twist spinning, Geom. Topol., Volume 10 (2006), pp. 27-56 | DOI | MR | Zbl

[36] Kim, Hee Jung; Ruberman, Daniel Smooth surfaces with non-simply-connected complements, Algebr. Geom. Topol., Volume 8 (2008) no. 4, pp. 2263-2287 | DOI | MR | Zbl

[37] Kim, Hee Jung; Ruberman, Daniel Topological triviality of smoothly knotted surfaces in 4-manifolds, Trans. Am. Math. Soc., Volume 360 (2008) no. 11, pp. 5869-5881 | DOI | MR | Zbl

[38] Kotschick, Dieter The Seiberg–Witten invariants of symplectic four-manifolds, Séminaire Bourbaki : volume 1995/96, exposés 805-819 (Astérisque), Volume 241, Société Mathématique de France, 1997, pp. 195-220 (talk:812) | Numdam | MR | Zbl

[39] Luttinger, Karl Murad Lagrangian tori in $R^{4}$ , J. Differ. Geom., Volume 42 (1995) no. 2, pp. 220-228 | MR | Zbl

[40] Mark, Thomas E. Knotted surfaces in 4-manifolds, Forum Math., Volume 25 (2013) no. 3, pp. 597-637 | DOI | MR | Zbl

[41] Moishezon, Boris Complex surfaces and connected sums of complex projective planes, Lecture Notes in Mathematics, 603, Springer, 1977, i+234 pages | DOI | MR | Zbl

[42] Morgan, John W.; Szabó, Zoltán; Taubes, Clifford Henry A product formula for the Seiberg–Witten invariants and the generalized Thom conjecture, J. Differ. Geom., Volume 44 (1996) no. 4, pp. 706-788 | MR | Zbl

[43] Oba, Takahiro Surfaces in the 4-disk with the same boundary and fundamental group, Math. Res. Lett., Volume 27 (2020) no. 1, pp. 265-279 | DOI | MR | Zbl

[44] Perron, B. Pseudo-isotopies et isotopies en dimension quatre dans la catégorie topologique, Topology, Volume 25 (1986) no. 4, pp. 381-397 | DOI | MR | Zbl

[45] Quinn, Frank Isotopy of $4$ -manifolds, J. Differ. Geom., Volume 24 (1986) no. 3, pp. 343-372 | DOI | MR | Zbl

[46] Ruberman, Daniel An obstruction to smooth isotopy in dimension $4$ , Math. Res. Lett., Volume 5 (1998) no. 6, pp. 743-758 | DOI | MR | Zbl

[47] Stong, Richard; Wang, Zhenghan Self-homeomorphisms of 4-manifolds with fundamental group Z, Topology Appl., Volume 106 (2000) no. 1, pp. 49-56 | DOI | MR | Zbl

[48] Sunukjian, Nathan S. Surfaces in 4-manifolds: concordance, isotopy, and surgery, Int. Math. Res. Not., Volume 2015 (2015) no. 17, pp. 7950-7978 | DOI | MR | Zbl

[49] Szabó, Zoltán Simply-connected irreducible $4$ -manifolds with no symplectic structures, Invent. Math., Volume 132 (1998) no. 3, pp. 457-466 | DOI | MR | Zbl

[50] Taubes, Clifford Henry The Seiberg–Witten invariants and symplectic forms, Math. Res. Lett., Volume 1 (1994) no. 6, pp. 809-822 | DOI | MR | Zbl

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