Monopoles and Landau-Ginzburg Models III: A Gluing Theorem

This is the third paper of this series. In \cite{Wang20}, we defined the monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is a compact oriented 3-manifold with toroidal boundary and $\omega$ 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 $\partial Y$ is disconnected, and $\omega$ is small and yet non-vanishing on $\partial Y$. 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 $H_2(Y,\partial Y;\mathbb{R})$ 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.