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GEOGRAPHY OF SPIN SYMPLECTIC FOUR-MANIFOLDS WITH ABELIAN FUNDAMENTAL GROUP

Part of: Differential topology

Published online by Cambridge University Press:  14 October 2011

RAFAEL TORRES
RAFAEL TORRES*
Affiliation:
Department of Mathematics, California Institute of Technology, 1200 E California Blvd, 91125 Pasadena CA, USA (email: [email protected])
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Abstract

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We study the geography and botany of symplectic spin four-manifolds with abelian fundamental group. By building on the constructions of J. Park and of B. D. Park and Szabó, we can give alternative proofs and extend several results on the geography of simply connected four-manifolds to the nonsimply connected realm.

Keywords

symplectic geometrygeography problemtorus surgeryexotic smooth structures

MSC classification

Secondary: 57R55: Differentiable structures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 91 , Issue 2 , October 2011 , pp. 207 - 218
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Akhmedov, A., Baldridge, S., Baykur, R. I., Kirk, P. and Park, B. D., ‘Simply connected minimal symplectic four-manifolds with signature less than −1’, J. Eur. Math. Soc. (JEMS) 1 (2010), 133–161.Google Scholar
[2]Akhmedov, A. and Park, B. D., ‘Exotic smooth structures on small four-manifolds with odd signature’, Invent. Math. 181 (2010), 577–603.CrossRefGoogle Scholar
[3]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.CrossRefGoogle Scholar
[4]Baldridge, S. and Kirk, P., ‘Constructions of small symplectic four-manifolds using Luttinger surgery’, J. Differential Geom. 82 (2009), 317–362.CrossRefGoogle Scholar
[5]Fintushel, R., Park, B. D. and Stern, R., ‘Reverse-engineering small four-manifolds’, Algebr. Geom. Topol. 7 (2007), 2103–2116.CrossRefGoogle Scholar
[6]Fintushel, R. and Stern, R., ‘Surgery in cusp neighborhoods and the geography of irreducible four-manifolds’, Invent. Math. 117 (1994), 455–523.CrossRefGoogle Scholar
[7]Fintushel, R. and Stern, R., ‘A fake four-manifold with π 1=ℤ’, Turkish J. Math. 18 (1994), 1–6.Google Scholar
[8]Fintushel, R. and Stern, R., ‘Knots, links and four-manifolds’, Invent. Math. 134 (1998), 363–400.CrossRefGoogle Scholar
[9]Fintushel, R. and Stern, R., ‘Surgery on nullhomologous tori and simply connected four-manifolds with b +2=1’, J. Topol. 1 (2008), 1–15.CrossRefGoogle Scholar
[10]Freedman, M., ‘The topology of four-dimensional manifolds’, J. Differential Geom. 17 (1982), 357–453.CrossRefGoogle Scholar
[11]Gompf, R. E., ‘Nuclei of elliptic surfaces’, Topology 30 (1991), 479–511.CrossRefGoogle Scholar
[12]Gompf, R. E., ‘A new construction of symplectic manifolds’, Ann. of Math. (2) 142 (1995), 527–595.CrossRefGoogle Scholar
[13]Gompf, R. E. and Stipsicz, A. I., 4-Manifolds and Kirby Calculus, Graduate Studies in Mathematics, 20 (American Mathematical Society, Providence, RI, 1999).CrossRefGoogle Scholar
[14]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
[15]Hambleton, I. and Teichner, P., ‘A non-extended hermitian form over ℤ[ℤ]’, Manuscripta Math. 94 (1997), 435–442.CrossRefGoogle Scholar
[16]Hamilton, M. J. D. and Kotshick, D., ‘Minimality and irreducibility of symplectic four-manifolds’, Int. Math. Res. Not. IMRN 2006 (2006), article ID 35032, 13 pp.Google Scholar
[17]Luttinger, K. M., ‘Lagrangian Tori in ℝ4’, J. Differential Geom. 42 (1995), 220–228.CrossRefGoogle Scholar
[18]McCarthy, J. and Wolfson, J., ‘Symplectic normal connect sum’, Topology 33 (1994), 729–764.CrossRefGoogle Scholar
[19]Morgan, J., Mrowka, M. and Szabó, Z., ‘Product formulas along T 3 for Seiberg–Witten invariants’, Math. Res. Lett. 4 (1997), 915–929.CrossRefGoogle Scholar
[20]Park, J., ‘The geography of irreducible four-manifolds’, Proc. Amer. Math. Soc. 126 (1998), 2493–2503.CrossRefGoogle Scholar
[21]Park, J., ‘The geography of spin symplectic four-manifolds’, Math. Z. 240 (2002), 405–421.CrossRefGoogle Scholar
[22]Park, B. D. and Szabó, Z., ‘The geography problem for irreducible spin four-manifolds’, Trans. Amer. Math. Soc. 352 (2000), 3639–3650.CrossRefGoogle Scholar
[23]Persson, U., Peters, C. and Xiao, G., ‘Geography of spin surfaces’, Topology 35 (1996), 845–862.CrossRefGoogle Scholar
[24]Stipsicz, A., ‘A note on the geography of symplectic four-manifolds’, Turkish J. Math. 20 (1996), 135–139.Google Scholar
[25]Taubes, C. H., ‘The Seiberg–Witten and Gromov invariants’, Math. Res. Lett. 2 (1995), 221–238.CrossRefGoogle Scholar
[26]Taubes, C. H., Seiberg Witten and Gromov Invariants for Symplectic Four-manifolds, First International Press Lecture Series, 2 (International Press, Somerville, 2000).Google Scholar
[27]Torres, R., ‘Geography and botany of symplectic four-manifolds with cyclic fundamental group’, (2009), arXiv:0903.5503.Google Scholar
[28]Torres, R., ‘Irreducible four-manifolds with abelian non-cyclic fundamental group’, Topology Appl. 157 (2010), 831–838.CrossRefGoogle Scholar