Monopoles and Landau–Ginzburg models III: A gluing theorem

Abstract This is the third paper of this series. In Wang [Monopoles and Landau‐Ginzburg models II: Floer homology. arXiv:2005.04333, 2020], we defined the monopole Floer homology for any pair , where is a compact oriented 3‐manifold with toroidal boundary and 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 is disconnected, and is small and yet non‐vanishing on . 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 that is irreducible, this Floer homology detects the Thurston norm on and the fiberness of . Finally, we show that our construction recovers the monopole link Floer homology for any link inside a closed 3‐manifold.