Abstract
We introduce techniques to study the topology of Stein fillings of a given contact three-manifold \((Y,\xi )\) which are not negative definite. For example, given a \(\hbox {spin}^c\) rational homology sphere \((Y,{\mathfrak {s}})\) with \({\mathfrak {s}}\) self-conjugate such that the reduced monopole Floer homology group \({\textit{HM}}_{\bullet }(Y,{\mathfrak {s}})\) has dimension one, we show that any Stein filling which is not negative definite has \(b_2^+=1\) or 2, and \(b_2^-\) is determined in terms of the Frøyshov invariant. The proof of this uses \(\text {Pin}(2)\)-monopole Floer homology. More generally, we prove that analogous statements hold under certain assumptions on the contact invariant of \(\xi \) and its interaction with \(\text {Pin}(2)\)-symmetry. We also discuss consequences for finiteness questions about Stein fillings.
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References
Baldwin, J.A.: Heegaard Floer homology and genus one, one-boundary component open books. J. Topol. 1(4), 963–992 (2008)
Bauer, S.: Almost complex 4-manifolds with vanishing first Chern class. J. Differ. Geom. 79(1), 25–32 (2008)
İnanç Baykur, R., Van Horn-Morris, J.: Families of contact \(3\)-manifolds with arbitrarily large Stein fillings. J. Differ. Geom. 101(3), 423–465 (2015). (With an appendix by Samuel Lisi and Chris Wendl)
Cieliebak, K., Eliashberg, Y.: From Stein to Weinstein and back. American Mathematical Society Colloquium Publications, vol. 59. American Mathematical Society, Providence, RI (2012). (Symplectic geometry of affine complex manifolds)
Colin, V., Ghiggini, P., Honda, K.: Equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions. Proc. Natl. Acad. Sci. USA 108(20), 8100–8105 (2011)
Conway, J., Min, H.: Classification of tight contact structures on surgeries on the figure-eight knot, to appear in Geom. Topology (2019)
Dai, I.: On the \(\text{ Pin }(2)\)-equivariant monopole Floer homology of plumbed 3-manifolds. Michigan Math. J. 67(2), 423–447 (2018)
Donaldson, S.K., Kronheimer, P.B.: The Geometry of Four-Manifolds. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York (1990)
Echeverria, M.: Naturality of the contact invariant in monopole Floer homology under strong symplectic cobordisms, to appear in Algebraic Geom. Topology (2018)
Eliashberg, Y.: Filling by holomorphic discs and its applications. Geometry of low-dimensional manifolds. 2 (Durham, 1989), volume 151 of London Mathematical Society Lecture Note Series, pp. 45–67. Cambridge University Press, Cambridge (1990)
Gompf, Robert E., Stipsicz, András I.: \(4\)-Manifolds and Kirby Calculus. Graduate Studies in Mathematics, vol. 20. American Mathematical Society, Providence, RI (1999)
Hanselman, J., Kutluhan, C., Lidman, T.: A remark on the geography problem in Heegaard Floer homology. Proc. of Symposia in Pure Math. (102) 103–113
Karakurt, Ç.: Contact structures on plumbed 3-manifolds. Kyoto J. Math. 54(2), 271–294 (2014)
Kutluhan, C., Lee, Y.-J., Taubes, C.: HF=HM I : Heegaard Floer homology and Seiberg–Witten Floer homology. preprint (2011). https://arxiv.org/abs/1007.1979
Kronheimer, P., Mrowka, T.: Monopoles and Three-Manifolds. New Mathematical Monographs, vol. 10. Cambridge University Press, Cambridge (2007)
Kronheimer, P., Mrowka, T., Ozsváth, P., Szabó, Z.: Monopoles and lens space surgeries. Ann. Math. (2) 165(2), 457–546 (2007)
Li, T.-J.: Symplectic 4-manifolds with Kodaira dimension zero. J. Differ. Geom. 74(2), 321–352 (2006)
Lin, F.: Lectures on monopole Floer homology. In: Proceedings of the Gökova Geometry-Topology Conference 2015, pp. 39–80. Gökova Geometry/Topology Conference (GGT), Gökova, (2016)
Lin, F.: The surgery exact triangle in \(\text{ Pin }(2)\)-monopole Floer homology. Algebr. Geom. Topol. 17(5), 2915–2960 (2017)
Lin, F.: A Morse–Bott approach to monopole Floer homology and the triangulation conjecture. Mem. Am. Math. Soc. 255(1221), 162 (2018)
Lisca, P.: On symplectic fillings of lens spaces. Trans. Am. Math. Soc. 360(2), 765–799 (2008)
Lisca, P., Matić, G.: Tight contact structures and Seiberg–Witten invariants. Invent. Math. 129(3), 509–525 (1997)
Manolescu, C.: Pin(2)-equivariant Seiberg-Witten Floer homology and the triangulation conjecture. J. Am. Math. Soc. 29(1), 147–176 (2016)
Martelli, B.: An introduction to Geometric Topology. arXiv:1610.02592, (2016)
McDuff, D.: The structure of rational and ruled symplectic \(4\)-manifolds. J. Am. Math. Soc. 3(3), 679–712 (1990)
Morgan, J.W.: The Seiberg–Witten Equations and Applications to the Topology of Smooth Four-Manifolds. Mathematical Notes, vol. 44. Princeton University Press, Princeton (1996)
Ozsváth, P., Szabó, Z.: Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary. Adv. Math. 173(2), 179–261 (2003)
Ozbagci, B., Stipsicz, A.I.: Surgery on Contact 3-Manifolds and Stein Surfaces, volume 13 of Bolyai Society Mathematical Studies. Springer, Berlin (2004)
Ozsváth, P., Szabó, Z.: Holomorphic disks and genus bounds. Geom. Topol. 8, 311–334 (2004)
Plamenevskaya, O.: Contact structures with distinct Heegaard Floer invariants. Math. Res. Lett. 11(4), 547–561 (2004)
Stipsicz, A.I.: Gauge theory and Stein fillings of certain 3-manifolds. Turkish J. Math. 26(1), 115–130 (2002)
Stoffregen, M.: Pin(2)-equivariant seiberg-witten floer homology of seifert fibrations. Compositio Mathematica 156(2), 199–250 (2020)
Taubes, C.H.: Embedded contact homology and Seiberg–Witten Floer cohomology I. Geom. Topol. 14(5), 2497–2581 (2010)
Wendl, C.: Strongly fillable contact manifolds and \(J\)-holomorphic foliations. Duke Math. J. 151(3), 337–384 (2010)
Acknowledgements
The author would like to thank András Stipsicz for sharing his expertise on the subject. This work was partially funded by NSF grant DMS-1807242.
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