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Sutured Floer homology, fibrations, and taut depth one foliations


Authors: Irida Altman, Stefan Friedl and András Juhász
Journal: Trans. Amer. Math. Soc. 368 (2016), 6363-6389
MSC (2010): Primary 57M25, 57M27, 57R30
DOI: https://doi.org/10.1090/tran/6610
Published electronically: November 17, 2015
MathSciNet review: 3461037
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Abstract: For an oriented irreducible 3-manifold $ M$ with non-empty toroidal boundary, we describe how sutured Floer homology ( $ \mathit {SFH}$) can be used to determine all fibred classes in $ H^1(M)$. Furthermore, we show that the $ \mathit {SFH}$ of a balanced sutured manifold $ (M,\gamma )$ detects which classes in $ H^1(M)$ admit a taut depth one foliation such that the only compact leaves are the components of  $ R(\gamma )$. The latter had been proved earlier by the first author under the extra assumption that $ H_2(M)=0$. The main technical result is that we can obtain an extremal $ \operatorname {Spin}^c$-structure  $ \mathfrak{s}$ (i.e., one that is in a `corner' of the support of $ \mathit {SFH}$) via a nice and taut sutured manifold decomposition even when  $ H_2(M) \neq 0$, assuming the corresponding group $ SFH(M,\gamma ,\mathfrak{s})$ has non-trivial Euler characteristic.


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Additional Information

Irida Altman
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: irida.altman@gmail.com

Stefan Friedl
Affiliation: Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
Email: sfriedl@gmail.com

András Juhász
Affiliation: Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
Email: juhasza@maths.ox.ac.uk

DOI: https://doi.org/10.1090/tran/6610
Received by editor(s): December 9, 2013
Received by editor(s) in revised form: August 18, 2014
Published electronically: November 17, 2015
Additional Notes: The third author was supported by a Royal Society Research Fellowship and OTKA grant NK81203
Article copyright: © Copyright 2015 American Mathematical Society