Abstract
We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface \(\Sigma _g\), where g is at least 2. In particular, we prove a uniqueness theorem asserting that any Stein filling must be s-cobordant rel boundary to the disk cotangent bundle of \(\Sigma _g\). For exact fillings, we show that the rational homology agrees with that of the disk cotangent bundle, and that the integral homology takes on finitely many possible values, including that of \(DT^*\Sigma _g\): for example, if \(g-1\) is square-free, then any exact filling has the same integral homology and intersection form as \(DT^*\Sigma _g\).
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Acknowledgements
This work began at the Princeton Low-Dimensional Topology Workshop 2015, and we thank the participants for contributing to a productive environment. We thank Matt Day, John Etnyre, Dave Futer and Yo’av Rieck for helpful conversations, and the referee for providing useful comments. We are especially grateful to Ian Agol for pointing out to us that Proposition 4.8 should be true and explaining how it should follow from the RFRS condition. SS was supported by NSF grant DMS-1506157. JVHM was supported in part by Simons Foundation Grant No. 279342.
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Communicated by Jean-Yves Welschinger.
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Sivek, S., Van Horn-Morris, J. Fillings of unit cotangent bundles. Math. Ann. 368, 1063–1080 (2017). https://doi.org/10.1007/s00208-016-1500-4
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DOI: https://doi.org/10.1007/s00208-016-1500-4