Virtual Betti numbers and the symplectic Kodaira dimension of fibered 4-manifolds

Baykur, R.

doi:10.1090/S0002-9939-2014-12151-4

Virtual Betti numbers and the symplectic Kodaira dimension of fibered $4$-manifolds
HTML articles powered by AMS MathViewer

by R. İnanç Baykur PDF

Proc. Amer. Math. Soc. 142 (2014), 4377-4384 Request permission

Abstract:

We prove that if a closed oriented $4$-manifold $X$ fibers over a $2$- or $3$-dimensional manifold, in most cases all of its virtual Betti numbers are infinite. In turn, we show that a closed oriented $4$-manifold $X$ which is not a tower of torus bundles and fibering over a $2$- or $3$-dimensional manifold does not admit a torsion symplectic canonical class, nor is it of Kodaira dimension zero.

References

Ian Agol, The virtual Haken conjecture, Doc. Math. 18 (2013), 1045–1087. With an appendix by Agol, Daniel Groves, and Jason Manning. MR 3104553
R. I. Baykur and S. Friedl, “Virtually symplectic fibered $4$-manifolds”, Indiana University Mathematics Journal, to appear: www.iumj.indiana.edu/IUMJ/Preprints/5591.pdf
Stefan Bauer, Almost complex 4-manifolds with vanishing first Chern class, J. Differential Geom. 79 (2008), no. 1, 25–32. MR 2414748
Jonathan Bowden, The topology of symplectic circle bundles, Trans. Amer. Math. Soc. 361 (2009), no. 10, 5457–5468. MR 2515819, DOI 10.1090/S0002-9947-09-04721-7
Josef Dorfmeister and Weiyi Zhang, The Kodaira dimension of Lefschetz fibrations, Asian J. Math. 13 (2009), no. 3, 341–357. MR 2570443, DOI 10.4310/AJM.2009.v13.n3.a5
Stefan Friedl and Stefano Vidussi, Symplectic 4-manifolds with $K=0$ and the Lubotzky alternative, Math. Res. Lett. 18 (2011), no. 3, 513–519. MR 2802584, DOI 10.4310/MRL.2011.v18.n3.a11
Nikolai V. Ivanov, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, vol. 115, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by E. J. F. Primrose and revised by the author. MR 1195787, DOI 10.1090/mmono/115
Sadayoshi Kojima, Finite covers of $3$-manifolds containing essential surfaces of Euler characteristic $=0$, Proc. Amer. Math. Soc. 101 (1987), no. 4, 743–747. MR 911044, DOI 10.1090/S0002-9939-1987-0911044-9
Tian-Jun Li, Smoothly embedded spheres in symplectic $4$-manifolds, Proc. Amer. Math. Soc. 127 (1999), no. 2, 609–613. MR 1459135, DOI 10.1090/S0002-9939-99-04457-3
Tian-Jun Li, Symplectic Parshin-Arakelov inequality, Internat. Math. Res. Notices 18 (2000), 941–954. MR 1792283, DOI 10.1155/S1073792800000490
Tian-Jun Li, Symplectic 4-manifolds with Kodaira dimension zero, J. Differential Geom. 74 (2006), no. 2, 321–352. MR 2259057
Tian-Jun Li, Quaternionic bundles and Betti numbers of symplectic 4-manifolds with Kodaira dimension zero, Int. Math. Res. Not. , posted on (2006), Art. ID 37385, 28. MR 2264722, DOI 10.1155/IMRN/2006/37385
John Luecke, Finite covers of $3$-manifolds containing essential tori, Trans. Amer. Math. Soc. 310 (1988), no. 1, 381–391. MR 965759, DOI 10.1090/S0002-9947-1988-0965759-2
John D. McCarthy, On the asphericity of a symplectic $M^3\times S^1$, Proc. Amer. Math. Soc. 129 (2001), no. 1, 257–264. MR 1707526, DOI 10.1090/S0002-9939-00-05571-4
Shigeyuki Morita, Characteristic classes of surface bundles, Invent. Math. 90 (1987), no. 3, 551–577. MR 914849, DOI 10.1007/BF01389178
Dusa McDuff and Dietmar Salamon, A survey of symplectic $4$-manifolds with $b^{+}=1$, Turkish J. Math. 20 (1996), no. 1, 47–60. MR 1392662
Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401