THE MINIMAL GENUS PROBLEM—A QUARTER CENTURY OF PROGRESS

doi:10.1007/s10473-022-0604-z

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (6): 2257-2278.doi: 10.1007/s10473-022-0604-z

THE MINIMAL GENUS PROBLEM—A QUARTER CENTURY OF PROGRESS

Josef G. DORFMEISTER¹, Tian-Jun LI²

1. Department of Mathematics, North Dakota State University, Fargo, ND 58102, USA;
2. School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

收稿日期:2022-06-30 出版日期:2022-12-25 发布日期:2022-12-16
通讯作者: Tian-Jun LI, E-mail: tjli@math.umn.edu E-mail:tjli@math.umn.edu
基金资助:
The second author was supported by NSF (1611680).

THE MINIMAL GENUS PROBLEM—A QUARTER CENTURY OF PROGRESS

Josef G. DORFMEISTER¹, Tian-Jun LI²

1. Department of Mathematics, North Dakota State University, Fargo, ND 58102, USA;
2. School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

Received:2022-06-30 Online:2022-12-25 Published:2022-12-16
Contact: Tian-Jun LI, E-mail: tjli@math.umn.edu E-mail:tjli@math.umn.edu
Supported by:
The second author was supported by NSF (1611680).

摘要/Abstract

摘要： This paper gives a survey of the progress on the minimal genus problem since Lawson’s [26] survey.

关键词: four manifolds, minimal genus, adjunction-type inequality, circle sum

Abstract: This paper gives a survey of the progress on the minimal genus problem since Lawson’s [26] survey.

Key words: four manifolds, minimal genus, adjunction-type inequality, circle sum

中图分类号:

57N13

Josef G. DORFMEISTER, Tian-Jun LI. THE MINIMAL GENUS PROBLEM—A QUARTER CENTURY OF PROGRESS[J]. 数学物理学报(英文版), 2022, 42(6): 2257-2278.

Josef G. DORFMEISTER, Tian-Jun LI. THE MINIMAL GENUS PROBLEM—A QUARTER CENTURY OF PROGRESS[J]. Acta mathematica scientia,Series B, 2022, 42(6): 2257-2278.

[1] Dai Bo, Ho Chung-I. The Cohomological Genus and Symplectic Genus for 4-Manifolds of Rational and Ruled Types. Mathematics, 2021, 9: 1563
[2] Dai Bo, Ho Chung-I, Li Tian-Jun. Minimal genus for 4-manifolds with b⁺= 1. J Topol, 2016, 9(1): 5-26
[3] Dorfmeister Josef G. Reduced classes of negative self-intersection in rational symplectic manifolds. in preparation
[4] Dorfmeister Josef G. Minimal genus and simplified classes in rational manifolds. submitted
[5] Dorfmeister Josef G, Li Tian-Jun, Wu Weiwei. Stability and existence of surfaces in symplectic 4-manifolds with b⁺= 1. J Reine Angew Math, 2018, 742: 115-155
[6] Edmonds Allan L. Tori in certain aspherical four-manifolds. Proc Amer Math Soc, 1998, 126(4): 1253-1255
[7] Friedman Robert, Morgan John W. On the diffeomorphism types of certain algebraic surfaces. II. J Differential Geom, 1988, 27(3): 371-398
[8] Friedl Stefan, Vidussi Stefano. Minimal genus in 4-manifolds with a free circle action. Adv Math, 2014, 250: 570-587
[9] Furuta M, Kametani Y, Minami N. Stable-homotopy Seiberg-Witten invariants for rational cohomology K³#K³’s. J Math Sci Univ Tokyo, 2001, 8(1): 157-176
[10] Furuta Mikio, Kametani Yukio, Matsue Hirofumi, Minami Norihiko. Homotopy theoretical considerations of the Bauer-Furuta stable homotopy Seiberg-Witten invariants//Proceedings of the Nishida Fest (Kinosaki 2003). Geom Topol Monogr, 10. Coventry: Geom Topol Publ, 2007: 155-166
[11] Gompf Robert E, Stipsicz András I. 4-manifolds and Kirby Calculus. Graduate Studies in Mathematics, 20. Providence, RI: American Mathematical Society, 1999
[12] Gompf Robert E. Minimal genera of open 4-manifolds. Geom Topol, 2017, 21(1): 107-155
[13] Hamilton M J D. Homology classes of negative square and embedded surfaces in 4-manifolds. Bull Lond Math Soc, 2013, 45(6): 1221-1226
[14] Hamilton M J D. The minimal genus problem for elliptic surfaces. Israel J Math, 2014, 200(1): 127-140
[15] Iida N, Mukherjee A, Taniguchi M. An adjunction inequality for the Bauer-Furuta type invariants, with applications to sliceness and 4-manifold topology. arXiv:2102.02076
[16] Kac Victor G. Infinite-Dimensional Lie Algebras. 3rd ed. Cambridge: Cambridge University Press, 1990
[17] Kikuchi Kazunori. Positive 2-spheres in 4-manifolds of signature (1, n). Pacific J Math, 1993, 160(2): 245- 258
[18] Kitayama Takahiro. A survey of the Thurston norm. arXiv:2108.08645
[19] Konno Hokuto. Bounds on genus and configurations of embedded surfaces in 4-manifolds. J Topol, 2016, 9(4): 1130-1152
[20] Kronheimer P B. Embedded surfaces and gauge theory in three and four dimensions//Surveys in differential geometry, Vol III (Cambridge, MA, 1996). Boston, MA: Int Press, 1998: 243-298
[21] Kronheimer P B. Minimal genus in S¹×M³. Invent Math, 1999, 135(1): 45-61
[22] Lambert-Cole Peter. Bridge trisections in CP² and the Thom conjecture. Geom Topol, 2020, 24(3): 1571- 1614
[23] Lambert-Cole Peter. Symplectic trisections and the adjunction inequality. arXiv:2009.11263
[24] Lambert-Cole Peter. An adjunction criterion in almost-complex 4-manifolds. arXiv:2107.10929
[25] Lawson Terry. Smooth embeddings of 2-spheres in 4-manifolds. Exposition Math, 1992, 10(4): 289-309
[26] Lawson Terry. The minimal genus problem. Exposition Math, 1997, 15(5): 385-431
[27] Lee Ronnie, Wilczyński Dariusz M. Representing homology classes by locally flat surfaces of minimum genus. Amer J Math, 1997, 119(5): 1119-1137
[28] Lee Ronnie, Wilczyński Dariusz M. Genus inequalities and four-dimensional surgery. Topology, 2000, 39(2): 311-330
[29] Levine Adam Simon, Ruberman Daniel, Strle Sašo. Nonorientable surfaces in homology cobordisms. With an appendix by Ira M Gessel. Geom Topol, 2015, 19(1): 439-494
[30] Li Bang-He. Generalization of the Whitney-Mahowald theorem. Trans Amer Math Soc, 1994, 346(2): 511-521
[31] Li Bang-He. Representing nonnegative homology classes of CP²#nCP² by minimal genus smooth embeddings. Trans Amer Math Soc, 2000, 352(9): 4155-4169
[32] Li Bang-He, Li Tian-Jun. Minimal genus smooth embeddings in S²×S² and CP²#nCP² with n ≤ 8. Topology, 1998, 37(3): 575-594
[33] Li Bang-He, Li Tian-Jun. Symplectic genus, minimal genus and diffeomorphisms. Asian J Math, 2002, 6(1): 123-144
[34] Li Bang-He, Li Tian-Jun. Smooth minimal genera for small negative classes in CP²#nCP² with n ≤ 9. Topology Appl, 2003, 132(1): 1-15
[35] Li Bang-He, Li Tian-Jun. On the diffeomorphism groups of rational and ruled 4-manifolds. J Math Kyoto Univ, 2006, 46(3): 583-593
[36] Li Bang-He, Li Tian-Jun. Circle-sum and minimal genus surfaces in ruled 4-manifolds. Proc Amer Math Soc, 2007, 135(11): 3745-3753
[37] Li Tian-Jun. Existence of symplectic surfaces. Geometry and topology of manifolds//Fields Inst Commun, 47. Providence, RI: Amer Math Soc, 2005: 203-217
[38] Li T J, Liu A. Symplectic structure on ruled surfaces and a generalized adjunction formula. Math Res Lett, 1995, 2(4): 453-471
[39] Lönne Michael. On the diffeomorphism groups of elliptic surfaces. Math Ann, 1998, 310(1): 103-117
[40] Nagel Matthias. Minimal genus in circle bundles over 3-manifolds. J Topol, 2016, 9(3): 747-773
[41] Nakashima Reito. Minimal genus problem for T²-bundles over surfaces. Algebr Geom Topol, 2021, 21(2): 893-916
[42] Nouh Mohamed Ait. The minimal genus problem in CP²#CP². Algebr Geom Topol, 2014, 14(2): 671-686
[43] Ozsváth Peter, Szabó Zoltán. The symplectic Thom conjecture. Ann of Math, 2000, 151(1): 93-124
[44] Ozsváth Peter, Szabó Zoltán. Higher type adjunction inequalities in Seiberg-Witten theory. J Differential Geom, 2000, 55(3): 385-440
[45] Ozsváth Peter, Szabó Zoltán. Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary. Adv Math, 2003, 173(2): 179-261
[46] Rohlin V A. Two-dimensional submanifolds of four-dimensional manifolds. (Russian) Funkcional Anal i Priložen, 1971, 5(1): 48-60
[47] Strle Sašo. Bounds on genus and geometric intersections from cylindrical end moduli spaces. J Differential Geom, 2003, 65(3): 469-511
[48] Wall C T C. On the orthogonal groups of unimodular quadratic forms. II. J Reine Angew Math, 1963/64, 213: 122-136
[49] Wall C T C. Diffeomorphisms of 4-manifolds. J London Math Soc, 1964, 39: 131-140
[50] Whitney Hassler. On the topology of differentiable manifolds//Lectures in Topology. University of Michigan Press, 1941: 101-141
[51] Wu Zhongtao. On minimal genus problem//Proceedings of the Gökova Geometry-Topology Conference 2014. Gökova Geometry/Topology Conference (GGT), Gökova, 2015: 250-263
[52] Zhao Xu’an, Gao Hongzhu, Qiu Huaidong. The minimal genus problem in rational surfaces CP²#nCP². Sci China Ser A, 2006, 49(9): 1275-1283
[53] Zhao Xu-An, Gao Hongzhu. The minimal genus problem in ruled manifolds. J Knot Theory Ramifications, 2008, 17(4): 471-482

THE MINIMAL GENUS PROBLEM—A QUARTER CENTURY OF PROGRESS

THE MINIMAL GENUS PROBLEM—A QUARTER CENTURY OF PROGRESS

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