In this paper, we study minimal simply connected 4-manifolds with b_2^+=3 which admit genus-4 Lefschetz fibrations over the 2-sphere. We first explicitly construct a genus-4 Lefschetz fibration over the 2-sphere using the monodromy of generalized Matsumoto fibration of genus 3 and the monodromy of the smallest genus-2 fibration given by Baykur and Korkmaz. We then construct two genus-4 Lefschetz fibrations over the 2-sphere that are exotic minimal symplectic 4-manifolds belonging to the homeomorphism classes of 3CP^2#15¯(CP^2 ) and 3CP^2#14¯(CP^2 ) by performing the fiber sum operation and then lantern substitution.
[1] Akhmedov A., Baykur R.İ., Park D. Constructing infinitely many smooth structures on small 4 -manifolds, J. Topol. 1 (2) (2008) 409-428.
[2] Akhmedov A, Park D., Exotic smooth structures on small 4 -manifolds with odd signatures, Invent. Math., 181(3) (2010) 577-603.
[3] Akhmedov A, Monden N., Genus-2 Lefschetz fibrations with b_2^+=1 and c_1^2=1,2, Kyoto J. Math., 60(4) (2020) 1419-1451.
[4] Altunöz T., Genus-3 Lefschetz fibrations and exotic 4-manifolds with b_2^+=3, Mediterr. J. Math., 18(3) (2021), Paper No. 102, 31.
[5] Altunöz T., Small genus-4 Lefschetz fibrations on simply-connected 4-manifolds, Turk. J. Math., 46(4) (2022), 1268-1290.
[6] Baldridge S., Kirk P., A symplectic manifold homeomorphic but not diffeomorphic to CP^2#3¯(CP^2 ) , Geom. Topol., 12 (2) (2008) 919-940.
[7] Baykur R.İ., Hamada N., Lefschetz fibrations with arbitrary signature, J. Eur. Math. Soc., 26 (2024) 2837-2895.
[8] Baykur R.İ., Hamada N., A small exotic symplectic rational surface, in preparation.
[9] Baykur R.İ., Small symplectic Calabi-Yau surfaces and exotic 4-manifolds via genus-3 pencils. In: Gauge theory and low-dimensional topology: progress and interaction, Open Book Series 5, (2002) 185-221.
[10] Baykur R.İ., Korkmaz M., Small Lefschetz fibrations and exotic 4-manifolds, Math. Ann., 367(3-4) (2017) 1333-1361.
[11] Baykur R.İ., Korkmaz M., Simone J., Geography of symplectic Lefschetz fibrations and rational blowdowns, Trans. Am.Math.Soc., 377(10) (2024) 6771-6792.
[12] Cadavid C., A remarkable set of words in the mapping class group, PhD, University of Texas, Austin, 1998.
[13] Donaldson S.K., Irrationality and the h -cobordism conjecture, J. Differential Geom., 26(1) (1987) 141-168.
[14] Dorfmeister J., Minimality of symplectic fiber sums along spheres, Asian J. Math., 17(3) (2013), 423–442.
[15] Endo H., Gurtas Y., Lantern relations and rational blowdowns, Proc. Amer. Math. Soc., 138(3) (2010) 1131-1142.
[16] Endo H., Nagami S., Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations, Trans. Am. Math. Soc., 357 (8) (2005) 3179-3199.
[17] Fintushel R., Park B.D., Stern R., Reverse engineering small 4-manifolds, Algebr. Geom. Topol., 7 (2007) 2103-2116.
[18] Fintushel R., Stern R., Rational blowdowns of smooth 4 -manifolds, J. Differ.l Geom., 46(2) (1997) 181-235.
[19] Fintushel R., Stern R., Double node neighborhoods and families of simply connected 4 -manifolds with b_2^+=1, J. Amer. Math. Soc. 19 (2006) 171–180
[20] Fintushel R., Stern R., Knots, links and 4-manifold, Invent. Math., 134(2) (1998) 363-400.
[21] Fintushel R., Stern R., Pinwheels and nullhomologous surgery on 4-manifolds with b_2^+=1, Algebr. Geom. Topol., 11(3) (2011) 1649-1699.
[22] Friedman R., Morgan J.W., Smooth four-manifolds and complex surfaces. Berlin Heidelberg New York, Springer. (1994).
[23] Gompf R.E., A new construction of symplectic manifolds, Ann. Math. Second Series, 142(3) (1995) 527-595.
[24] Hamada N., Hayano K., Topology of holomorphic Lefschetz pencils on the four-torus, Algebr. Geom. Topol. 18(3) (2018) 1515–1572.
[25]Korkmaz M., Noncomplex smooth 4-manifolds with Lefschetz fibrations, Int. Math. Res. Not., 3 (2001) 115-128.
[26] Koschick D., On manifolds homeomorphic to CP^2#8¯(CP^2 ), Invent. Math., 95(3) (1989) 591-600.
[27] Matsumoto Y., Lefschetz fibrations of genus two- a topological approach, Topology and Teichmüller spaces, (1996) 123-148.
[28] Park B.D., Exotic smooth structures on 3CP^2#n¯(CP^2 ), Proc. Amer. Math. Soc., 128(10) (2000) 3057–3065.
[29] Park B.D., Exotic smooth structures on 3CP^2#n¯(CP^2 ), Part II, Proc. Amer. Math. Soc., 128(10) (2000) 3067–3073.
[30] Park B.D., Constructing infinitely many smooth structures on 3CP^2#n¯(CP^2 ), Math. Ann., 322(2) (2002) 267–278.
[31] Park J., Simply connected symplectic 4-manifolds with b_2^+=1 and c_1^2=2, Invent. Math., 159(3) (2005) 657-667.
[32] Park J., Stipsicz A., Szabó Z., Exotic smooth structures on CP^2#5¯(CP^2 ), Math. Res. Lett. 12(5-6) (2005) 701-712.
[33] Park J., Stipsicz A., Szabó Z., An exotic smooth structure on CP^2#6¯(CP^2 ), Geom. Topol.,. 9 (2005) 813-832.
[34] Stipsicz A., Donaldson invariants of certain symplectic manifolds, J. Reine Angew. Math. 465 (1995) 1-10.
[35] Stipsicz A., Szabó Z., The smooth classification of elliptic surfaces with b^+>1, Duke Math. J., 75(1) (1994) 1-50.
[36] Stipsicz A., Szabó Z., Small exotic 4-manifolds with b^+=3, Bull. London Math. Soc., 38(3) (2006) 501-506.
[37] Stipsicz A., Yun K-Y., On minimal number of singular fibers in Lefschetz fibrations over the torus, Proc. Amer. Math. Soc. 145(8) (2017) 3607-3616.
[38] Szabó Z., Irreducible four-manifolds with small Euler characteristics, Topology, 35(2) (1996) 411-426.
[39] Usher M., Minimality and Symplectic Sums, Int. Math. Res. Not. Art ID 49857, (2006), 17.
Year 2024,
Volume: 45 Issue: 4, 796 - 802, 30.12.2024
[1] Akhmedov A., Baykur R.İ., Park D. Constructing infinitely many smooth structures on small 4 -manifolds, J. Topol. 1 (2) (2008) 409-428.
[2] Akhmedov A, Park D., Exotic smooth structures on small 4 -manifolds with odd signatures, Invent. Math., 181(3) (2010) 577-603.
[3] Akhmedov A, Monden N., Genus-2 Lefschetz fibrations with b_2^+=1 and c_1^2=1,2, Kyoto J. Math., 60(4) (2020) 1419-1451.
[4] Altunöz T., Genus-3 Lefschetz fibrations and exotic 4-manifolds with b_2^+=3, Mediterr. J. Math., 18(3) (2021), Paper No. 102, 31.
[5] Altunöz T., Small genus-4 Lefschetz fibrations on simply-connected 4-manifolds, Turk. J. Math., 46(4) (2022), 1268-1290.
[6] Baldridge S., Kirk P., A symplectic manifold homeomorphic but not diffeomorphic to CP^2#3¯(CP^2 ) , Geom. Topol., 12 (2) (2008) 919-940.
[7] Baykur R.İ., Hamada N., Lefschetz fibrations with arbitrary signature, J. Eur. Math. Soc., 26 (2024) 2837-2895.
[8] Baykur R.İ., Hamada N., A small exotic symplectic rational surface, in preparation.
[9] Baykur R.İ., Small symplectic Calabi-Yau surfaces and exotic 4-manifolds via genus-3 pencils. In: Gauge theory and low-dimensional topology: progress and interaction, Open Book Series 5, (2002) 185-221.
[10] Baykur R.İ., Korkmaz M., Small Lefschetz fibrations and exotic 4-manifolds, Math. Ann., 367(3-4) (2017) 1333-1361.
[11] Baykur R.İ., Korkmaz M., Simone J., Geography of symplectic Lefschetz fibrations and rational blowdowns, Trans. Am.Math.Soc., 377(10) (2024) 6771-6792.
[12] Cadavid C., A remarkable set of words in the mapping class group, PhD, University of Texas, Austin, 1998.
[13] Donaldson S.K., Irrationality and the h -cobordism conjecture, J. Differential Geom., 26(1) (1987) 141-168.
[14] Dorfmeister J., Minimality of symplectic fiber sums along spheres, Asian J. Math., 17(3) (2013), 423–442.
[15] Endo H., Gurtas Y., Lantern relations and rational blowdowns, Proc. Amer. Math. Soc., 138(3) (2010) 1131-1142.
[16] Endo H., Nagami S., Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations, Trans. Am. Math. Soc., 357 (8) (2005) 3179-3199.
[17] Fintushel R., Park B.D., Stern R., Reverse engineering small 4-manifolds, Algebr. Geom. Topol., 7 (2007) 2103-2116.
[18] Fintushel R., Stern R., Rational blowdowns of smooth 4 -manifolds, J. Differ.l Geom., 46(2) (1997) 181-235.
[19] Fintushel R., Stern R., Double node neighborhoods and families of simply connected 4 -manifolds with b_2^+=1, J. Amer. Math. Soc. 19 (2006) 171–180
[20] Fintushel R., Stern R., Knots, links and 4-manifold, Invent. Math., 134(2) (1998) 363-400.
[21] Fintushel R., Stern R., Pinwheels and nullhomologous surgery on 4-manifolds with b_2^+=1, Algebr. Geom. Topol., 11(3) (2011) 1649-1699.
[22] Friedman R., Morgan J.W., Smooth four-manifolds and complex surfaces. Berlin Heidelberg New York, Springer. (1994).
[23] Gompf R.E., A new construction of symplectic manifolds, Ann. Math. Second Series, 142(3) (1995) 527-595.
[24] Hamada N., Hayano K., Topology of holomorphic Lefschetz pencils on the four-torus, Algebr. Geom. Topol. 18(3) (2018) 1515–1572.
[25]Korkmaz M., Noncomplex smooth 4-manifolds with Lefschetz fibrations, Int. Math. Res. Not., 3 (2001) 115-128.
[26] Koschick D., On manifolds homeomorphic to CP^2#8¯(CP^2 ), Invent. Math., 95(3) (1989) 591-600.
[27] Matsumoto Y., Lefschetz fibrations of genus two- a topological approach, Topology and Teichmüller spaces, (1996) 123-148.
[28] Park B.D., Exotic smooth structures on 3CP^2#n¯(CP^2 ), Proc. Amer. Math. Soc., 128(10) (2000) 3057–3065.
[29] Park B.D., Exotic smooth structures on 3CP^2#n¯(CP^2 ), Part II, Proc. Amer. Math. Soc., 128(10) (2000) 3067–3073.
[30] Park B.D., Constructing infinitely many smooth structures on 3CP^2#n¯(CP^2 ), Math. Ann., 322(2) (2002) 267–278.
[31] Park J., Simply connected symplectic 4-manifolds with b_2^+=1 and c_1^2=2, Invent. Math., 159(3) (2005) 657-667.
[32] Park J., Stipsicz A., Szabó Z., Exotic smooth structures on CP^2#5¯(CP^2 ), Math. Res. Lett. 12(5-6) (2005) 701-712.
[33] Park J., Stipsicz A., Szabó Z., An exotic smooth structure on CP^2#6¯(CP^2 ), Geom. Topol.,. 9 (2005) 813-832.
[34] Stipsicz A., Donaldson invariants of certain symplectic manifolds, J. Reine Angew. Math. 465 (1995) 1-10.
[35] Stipsicz A., Szabó Z., The smooth classification of elliptic surfaces with b^+>1, Duke Math. J., 75(1) (1994) 1-50.
[36] Stipsicz A., Szabó Z., Small exotic 4-manifolds with b^+=3, Bull. London Math. Soc., 38(3) (2006) 501-506.
[37] Stipsicz A., Yun K-Y., On minimal number of singular fibers in Lefschetz fibrations over the torus, Proc. Amer. Math. Soc. 145(8) (2017) 3607-3616.
[38] Szabó Z., Irreducible four-manifolds with small Euler characteristics, Topology, 35(2) (1996) 411-426.
[39] Usher M., Minimality and Symplectic Sums, Int. Math. Res. Not. Art ID 49857, (2006), 17.