2020/10/09 by Donghao Wang, Wang, Donghao
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.2010.04318
This is the third paper of this series. In \citeWang20, we defined the monopole Floer homology for any pair (Y,ω), where Y is a compact oriented 3-manifold with toroidal boundary and ω is a suitable closed 2-form viewed as a decoration. In this paper, we establish a gluing theorem for this Floer homology when two such 3-manifolds are glued suitably along their common boundary, assuming that ∂ Y is disconnected, and ω is small and yet non-vanishing on ∂ Y. As applications, we construct a monopole Floer 2-functor and the generalized cobordism maps. Using results of Kronheimer-Mrowka and Ni, it is shown that for any such 3-manifold Y that is irreducible, this Floer homology detects the Thurston norm on H2(Y,∂ Y;ℝ) and the fiberness of Y. Finally, we show that our construction recovers the monopole link Floer homology for any link inside a closed 3-manifold.