GEOGRAPHY AND BOTANY OF IRREDUCIBLE NON-SPIN SYMPLECTIC 4-MANIFOLDS WITH ABELIAN FUNDAMENTAL GROUP
Glasgow mathematical journal, Tome 56 (2014) no. 2, pp. 261-281

Voir la notice de l'article provenant de la source Cambridge University Press

The geography and botany problems of irreducible non-spin symplectic 4-manifolds with a choice of fundamental group from $\{{\mathbb{Z}}_p, {\mathbb{Z}}_p\oplus {\mathbb{Z}}_q, {\mathbb{Z}}, {\mathbb{Z}}\oplus {\mathbb{Z}}_p, {\mathbb{Z}}\oplus {\mathbb{Z}}\}$ are studied by building upon the recent progress obtained on the simply connected realm. Results on the botany of simply connected 4-manifolds not available in the literature are extended.
DOI : 10.1017/S0017089513000232
Mots-clés : Primary 57R17, Secondary 57M05, 54D05
TORRES, RAFAEL. GEOGRAPHY AND BOTANY OF IRREDUCIBLE NON-SPIN SYMPLECTIC 4-MANIFOLDS WITH ABELIAN FUNDAMENTAL GROUP. Glasgow mathematical journal, Tome 56 (2014) no. 2, pp. 261-281. doi: 10.1017/S0017089513000232
@article{10_1017_S0017089513000232,
     author = {TORRES, RAFAEL},
     title = {GEOGRAPHY {AND} {BOTANY} {OF} {IRREDUCIBLE} {NON-SPIN} {SYMPLECTIC} {4-MANIFOLDS} {WITH} {ABELIAN} {FUNDAMENTAL} {GROUP}},
     journal = {Glasgow mathematical journal},
     pages = {261--281},
     year = {2014},
     volume = {56},
     number = {2},
     doi = {10.1017/S0017089513000232},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089513000232/}
}
TY  - JOUR
AU  - TORRES, RAFAEL
TI  - GEOGRAPHY AND BOTANY OF IRREDUCIBLE NON-SPIN SYMPLECTIC 4-MANIFOLDS WITH ABELIAN FUNDAMENTAL GROUP
JO  - Glasgow mathematical journal
PY  - 2014
SP  - 261
EP  - 281
VL  - 56
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089513000232/
DO  - 10.1017/S0017089513000232
ID  - 10_1017_S0017089513000232
ER  - 
%0 Journal Article
%A TORRES, RAFAEL
%T GEOGRAPHY AND BOTANY OF IRREDUCIBLE NON-SPIN SYMPLECTIC 4-MANIFOLDS WITH ABELIAN FUNDAMENTAL GROUP
%J Glasgow mathematical journal
%D 2014
%P 261-281
%V 56
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089513000232/
%R 10.1017/S0017089513000232
%F 10_1017_S0017089513000232

[1] 1.Akhmedov, A., Small exotic 4-manifolds, Alg. Geom. Topol. 8 (2008), 1781–1794. Google Scholar

[2] 2.Akhmedov, A., Baldridge, S., Baykur, R. I., Kirk, P. and Park, B. D., Simply connected minimal symplectic 4-manifolds with signature less than -1, J. Eur. Math. Soc. 1 (2010), 133–161. Google Scholar

[3] 3.Akhmedov, A., Baykur, R. I. and Park, B. D., Constructing infinitely many smooth structures on small 4-manifolds, J. Topol. 2 (2008), 1–13. Google Scholar

[4] 4.Akhmedov, A. and Park, B. D., Exotic smooth structures on small 4-manifolds, Invent. Math. 173 (2008), 209–223. Google Scholar

[5] 5.Akhmedov, A. and Park, B. D., New symplectic 4-manifolds with non-negative signature, J. Gokova Geom. Topol. 2 (2008), 1–13. Google Scholar

[6] 6.Akhmedov, A. and Park, B. D., Exotic smooth structures on small 4-manifolds with odd signatures, Invent. Math. 181 (2010), 577–603. Google Scholar

[7] 7.Akhmedov, A. and Park, B. D., Geography of simply connected spin symplectic 4-manifolds, Math. Res. Lett. 17 (2010), 483–492. Google Scholar

[8] 8.Auroux, D., Donaldson, S. K. and Katzarkov, L., Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves, Math. Ann. 326 (2003) 185–203. Google Scholar | DOI

[9] 9.Baldridge, S. and Kirk, P., Symplectic 4-manifolds with arbitrary fundamental group near the Bogomolov-Miyaoka-Yau line, J. Symplectic Geom. 4 (2006), 63–70. Google Scholar

[10] 10.Baldridge, S. and Kirk, P., On symplectic 4-manifolds with prescribed fundamental group, Comment. Math. Helv. 82 (2007), 845–875. Google Scholar

[11] 11.Baldridge, S. and Kirk, P., A symplectic manifold homeomorphic but not diffeomorphic to , Geom. Topol. 12 (2) (2008), 919–940. Google Scholar

[12] 12.Baldridge, S. and Kirk, P., Luttinger surgery and interesting symplectic 4-manifolds with small Euler characteristic, arXiv:math/0701400 . Google Scholar

[13] 13.Baldridge, S. and Kirk, P., Constructions of small symplectic 4-manifolds using Luttinger surgery, J. Diff. Geom. 82 (2) (2009), 317–362. Google Scholar

[14] 14.Fintushel, R. A., Knot surgery revisited, in Floer Homology, Gauge Theory and Low Dimensional Topology, Clay Mathematics Institute Proceddings 4, CMI/AMS Book Series (Clay Mathematics Institute, Oxford, UK, 2006), 195–224. Google Scholar

[15] 15.Fintushel, R., Construction of 4-manifolds, Talk given at the Conference of Perspectives in Analysis, Geometry, and Topology, Stockholm University, Stockholm, Sweden (2008). Google Scholar

[16] 16.Fintushel, R. A., Park, B. D. and Stern, R. J.Reverse engineering small 4-manifolds, Alg. Geom. Topol. 7 (2007), 2103–2116. Google Scholar

[17] 17.Fintushel, R. A. and Stern, R. J., A fake 4-manifold with , Turkish J. Math. 18 (1994), 1–6. Google Scholar

[18] 18.Fintushel, R. A. and Stern, R. J., Knots, links and 4-manifolds, Invent. Math. 134 (1998), 363–400. Google Scholar | DOI

[19] 19.Fintushel, R. A. and Stern, R. J., Six lectures on four 4-manifolds, Graduate Summer School on Low Dimensional Topology (Park City Mathematics Institute, Princeton, NJ, 2006). Google Scholar

[20] 20.Fintushel, R. A. and Stern, R. J., Surgery on nullhomologous tori and simply connected 4-manifolds with b + = 1, J. Topol. 1 (2008), 1–15. Google Scholar

[21] 21.Fintushel, R. A. and Stern, R. J., Pinwheels and nullhomologous surgery on 4-manifolds with b + = 1, Alg. Geom. Topol. 11 (2011), 1649–1699. Google Scholar | DOI

[22] 22.Fintushel, R. A., Stern, R. J. and Sunukjian, N., Exotic group actions on simply-connected 4-manifolds, J. Topol. 2 (2009), 769–822. Google Scholar

[23] 23.Freedman, M., The topology of four-dimensional manifolds, J. Diff. Geom. 17 (1982), 357–453. Google Scholar

[24] 24.Freedman, M. and Quinn, F., Topology of 4-manifolds (Princeton University Press, Princeton, NJ, 1990). Google Scholar

[25] 25.Gompf, R. E., A new construction of symplectic manifolds. Ann. Math. 142 (1995), 527–595. Google Scholar

[26] 26.Gompf, R. E. and Stipsicz, A. I., 4-manifolds and Kirby calculus, Graduate Studies in Mathematics, 20 (American Mathematical Society, Providence, RI, 1999). Google Scholar | DOI

[27] 27.Hambleton, I. and Kreck, M., Smooth structures on algebraic surfaces with cyclic fundamental group, Invent. Math. 91 (1988), 53–59. Google Scholar

[28] 28.Hambleton, I. and Kreck, M., Smooth structures on algebraic surfaces with finite fundamental group, Invent. Math. 102 (1990), 109–114. Google Scholar

[29] 29.Hambleton, I. and Kreck, M., Cancellation, elliptic surfaces and the topology of certain four-manifolds, J. Reine Angew. Math. 444 (1993), 79–100. Google Scholar

[30] 30.Hambleton, I. and Teichner, P., A non-extended Hermitian form over , Manuscripta Math. 94 (1997), 435–442. Google Scholar | DOI

[31] 31.Hamilton, M. J. D. and Kotschick, D., Minimality and irreducibility of symplectic four-manifolds, Int. Math. Res. Not. 13, art. ID 35032 (2006). doi:10.1155/IMRN/2006/35032. Google Scholar

[32] 32.Kotschick, D., The Seiberg-Witten invariants of symplectic 4-manifolds after C.H. Taubes}, Seminaire Bourbaki 48éme année 812 (1995–96), 195–220 (Astérique (1997)). Google Scholar

[33] 33.Krushkal, V. and Lee, R., Surgery on closed 4-manifolds with free fundamental group, Math. Proc. Camb. Phil. Soc. 133 (2) (2002), 305–310. Google Scholar | DOI

[34] 34.Lübke, M. and Okonek, C., Differentiable structures on elliptic surfaces with cyclic fundamental group, Comp. Math. 63 (1987), 217–222. Google Scholar

[35] 35.Luttinger, K. M., Lagrangian tori in , J. Diff. Geom. 42 (1995), 220–228. Google Scholar

[36] 36.Maier, F., On the diffeomorphism type of elliptic surfaces with finite fundamental group, PhD Thesis, Tulane University, New Orleans, LA (1987). Google Scholar

[37] 37.Mccarthy, J. and Wolfson, J., Symplectic normal connect sum, Topology 33 (1994), 729–764. Google Scholar

[38] 38.Mcduff, D. and Salamon, D., Introduction to symplectic topology, 2nd ed. Oxford Mathematical Monographs. (Clarendon Press, Oxford University Press, New York, 1998). Google Scholar

[39] 39.Morgan, J., Mrowka, T. and Szabó, Z., Product formulas along T 3 for Seiberg-Witten invariants, Math. Res. Lett. 2 (1997) 915–929. Google Scholar

[40] 40.Okonek, C., Fake Enrique surfaces, Topology 27 (1988), 415–427. Google Scholar

[41] 41.Ozbagci, B. and Stipsicz, A., Noncomplex smooth 4-manifolds with genus-2 Lefschetz fibrations, Proc. Amer. Math. Soc. 128 (10) (2000), 3125–3128. Google Scholar

[42] 42.Ozbagci, B. and Stipsicz, A., Surgery con contact 3-manifolds and Stein surfaces, Bolyai Society Mathematical Studies, vol 13 (Springer, New York, 2006). Google Scholar

[43] 43.Park, J., The geography of irreducible 4-manifolds, Proc. Amer. Math. Soc. 126 (1998), 2493–2503. Google Scholar

[44] 44.Park, B. D., Exotic smooth structures on Proc. Amer. Math. Soc. 128 (10) (2000), 3057–3065, and Erratum Proc. Amer. Math. Soc. (4), (2008), 1503. Google Scholar

[45] 45.Park, B. D., Exotic smooth structures on part II, Proc. Amer. Math. Soc. 128 (10) (2000), 3067–3073. Google Scholar

[46] 46.Park, B. D., A gluing formula for the Seiberg-Witten invariant along $T^3$, Michigan Math. J. 50 (2002), 593–611. Google Scholar

[47] 47.Park, B. D., Constructing infinitely many smooth structures on , Math. Ann. 322 (2) (2002), 267–278; and Erratum Math. Ann. (2008), 731–732. Google Scholar | DOI

[48] 48.Park, J., The geography of spin symplectic 4-manifolds, Math. Z. 240 (2002), 405–421. Google Scholar

[49] 49.Park, J., Exotic smooth structures on 4-manifolds, Forum Math. 14 (2002), 915–929. Google Scholar

[50] 50.Park, J., Exotic smooth structures on 4-manifolds II, Top. Appl. 132 (2003), 195–202. Google Scholar

[51] 51.Park, J., The geography of symplectic 4-manifolds with an arbitrary fundamental group, Proc. Amer. Math. Soc. 135 (7) (2007) 2301–2307. Google Scholar

[52] 52.Park, B. D. and Szabó, Z.. The geography problem for irreducible spin four-manifolds, Trans. Amer. Math. Soc. 352 (2000), 3639–3650. Google Scholar | DOI

[53] 53.Park, J. and Yun, K. H., Exotic smooth structures on , Bull. Korean Math. Soc. 47 (2010), 961–971. Google Scholar

[54] 54.Rokhlin, V., New results in the theory of four dimensional manifolds, Dokl. Akad. Nauk. USSR 84 (1951), 355–357. Google Scholar

[55] 55.Smith, I., Symplectic geometry of Lefschetz fibrations, Dissertation (Oxford Press University, Oxford, UK, 1998). Google Scholar

[56] 56.Stern, R. J., Topology of 4-manifolds: a conference in honor of Ronald Fintushel's 60th birthday, talk given at Tulane University (2006). Google Scholar

[57] 57.Stipsicz, A. I., A note on the geography of symplectic manifolds, Turkish J. Math. 20 (1996), 135–139. Google Scholar

[58] 58.Stipsicz, A. I., Simply-connected symplectic 4-manifolds with positive signature, proceedings of the 6th Gokova geometry-topology conference, Turkish J. Math. 23 (1999), 145–150. Google Scholar

[59] 59.Stipsicz, A. I., The geography problem of 4-manifolds with various structures, Acta Math. Hungar. 87 (2000), 267–278. Google Scholar

[60] 60.Stipsicz, A. I. and Szabó, S., An exotic smooth structure on , Geom. Topol. 9 (2005), 813–832. Google Scholar

[61] 61.Stipsicz, A. I. and Szabó, S., Small exotic 4-manifolds with b +2 = 3, Bull. Lond. Math. Soc. 38 (2006), 501–506. Google Scholar

[62] 62.Stong, R. and Wang, Z., Self-homeomorphisms of 4-manifolds with fundamental group , Top. Appl. 106 (2000), 49–56. Google Scholar

[63] 63.Szabó, Z., Irreducible four-manifolds with small Euler characteristics, Topology 35, (1996), 411–426. Google Scholar | DOI

[64] 64.Szabó, Z., Simply-connected irreducible 4-manifolds with no symplectic structures, Invent. Math. 132, (1998), 457–466. Google Scholar

[65] 65.Taubes, C., The Seiberg-Witten invariants and symplectic forms, Math. Res. Let. 1 (6) (1994), 809–822. Google Scholar

[66] 66.Taubes, C., Seiberg-Witten and Gromov invariants, in Geometry and Physics (Aarhus, 1995), 591–601, Lecture Notes in Pure and Applied Mathematics, 184, (Marcel Dekker New York, 1997). Google Scholar

[67] 67.Taubes, C., Counting pseudo-holomorphic submanifolds in dimension 4, J. Diff. Geom. 44 (1996), 818–893. Google Scholar

[68] 68.Thurston, W., Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), 467–468. Google Scholar

[69] 69.Torres, R., Irreducible 4-manifolds with abelian non-cyclic fundamental group of small rank, Top. Appl. 57 (2010), 831–838. Google Scholar

[70] 70.Torres, R., Geography of spin symplectic four-manifolds with abelian fundamental group, J. Aust. Math. Soc. 91 (2011), 207–218. Google Scholar | DOI

[71] 71.Usher, M., Minimality and symplectic sums, Int. Math. Res. Not. (2006), Art. ID 49857, 17. doi:10.1155/IMRN/2006/49857. Google Scholar | DOI

[72] 72.Wang, S., Smooth structures on complex surfaces with fundamental group 2, Proc. Amer. Math. Soc. 125 (1) (1997), 287–292. Google Scholar

[73] 73.Witten, E., Monopoles and four-manifolds, Math. Res. Lett. 1 (6) (1994), 769–796. Google Scholar

[74] 74.Yazinski, J. T., A new bound on the size of symplectic 4-manifolds with prescribed fundamental group, J. Symplectic Geom. 11 (2013), 25–36. Google Scholar | DOI

Cité par Sources :