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Exotic 4 -Manifolds from Genus-4 Lefschetz Fibrations

Year 2024, Volume: 45 Issue: 4, 796 - 802, 30.12.2024

Tülin Altunöz

https://doi.org/10.17776/csj.1520855

Abstract

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.

Keywords

Lefschetz fibrations Symplectic 4-manifolds Exotic manifolds Mapping class groups.

References

  • [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

Tülin Altunöz

https://doi.org/10.17776/csj.1520855

Abstract

References

  • [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.
There are 39 citations in total.

Details

Primary Language English
Subjects Topology
Journal Section Natural Sciences
Authors

Tülin Altunöz 0000-0002-9116-2253

Publication Date December 30, 2024
Submission Date July 23, 2024
Acceptance Date December 26, 2024
Published in Issue Year 2024Volume: 45 Issue: 4

Cite

APA Altunöz, T. (2024). Exotic 4 -Manifolds from Genus-4 Lefschetz Fibrations. Cumhuriyet Science Journal, 45(4), 796-802. https://doi.org/10.17776/csj.1520855
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