Resumen de Decomposing sutured monopole and instanton Floer homologies

Sudipta Ghosh, Zhenkun Li

• In this paper, we generalize the work of the second author in Li (Direct systems and the knot monopole Floer homology, 2019. arXiv:1901.06679) and prove a grading shifting property, in sutured monopole and instanton Floer theories, for general balanced sutured manifolds.

  This result has a few consequences. First, we offer an algorithm that computes the Floer homologies of a family of sutured handlebodies..

  Second, we obtain a canonical decomposition of sutured monopole and instanton Floer homologies and build polytopes for these two theories, which was initially achieved by Juhász (Geom Topol 14(3):1303–1354, 2010) for sutured (Heegaard) Floer homology.

  Third, we establish a Thurston-norm detection result for monopole and instanton knot Floer homologies, which were introduced by Kronheimer and Mrowka (J Differ Geom 84(2):301–364, 2010). The same result was originally proved by Ozsváth and Szabó for link Floer homology in Ozsváth and Szabó (J Am Math Soc 21(3):671–709, 2008). Last, we generalize the construction of minus versions of monopole and instanton knot Floer homology, which was initially done for knots by the second author in Li (2019), to the case of links. Along with the construction of polytopes, we also proved that, for a balanced sutured manifold with vanishing second homology, the rank of the sutured monopole or instanton Floer homology bounds the depth of the balanced sutured manifold. As a corollary, we obtain an independent proof that monopole and instanton knot Floer homologies, as mentioned above, both detect fibred knots in S3.

  This result was originally achieved by Kronheimer and Mrowka (2010).