Decomposing sutured monopole and Instanton Floer homologies

Sudipta Ghosh, Zhenkun Li

Published 2019-10-23Version 1

The Sutured monopole and instanton Floer homologies are defined by Kronheimer and Mrowka. In this paper, we prove that in sutured monopole Floer theory, on any closure of a balanced sutured manifold, the difference of any two supporting spin${}^c$ structures, in terms of their first Chern classes, is supported in the original sutured manifold. We also proved a similar result in the instanton settings. Using the above facts, we prove a general grading shifting formula in sutured monopole and instanton Floer theories, generalizing the work by the second author. Applying this grading shifting property, we obtain decompositions of sutured monopole and instanton Floer homologies in a canonical way, disregarding the choices of closures. As a result, we construct polytopes in sutured monopole and instanton Floer theories which resemble the one constructed by Juh\'asz for sutured (Heegaard) Floer homology. We prove that the polytopes must have maximal dimensions for some special classes of balanced sutured manifolds. In proving these results, we also show that sutured monopole and instanton Floer homologies can be used to bound the depth of balanced sutured manifolds with vanishing second homology. In particular, this yields an independent proof that the monopole and instanton knot Floer homologies constructed by Kronheimer and Mrowka both detect fibred knots in $S^3$. Using the general grading shifting formula, we also describe an algorithm to compute the sutured monopole or instanton Floer homology of some (families of) sutures on a handle body (of arbitrary genus). Moreover, we extend the construction of minus versions of knot Floer homology in monopole and instanton Floer theories, which were originally done by the second author for knots, to the case of links.