Pin(2)-monopole Floer homology, higher compositions and connected sums

We study the behavior of Pin (2)-monopole Floer homology under connected sums. After constructing a (partially defined) A∞-module structure on the Pin (2)-monopole Floer chain complex of a three-manifold (in the spirit of Baldwin and Bloom's monopole category), we identify up to quasi-isomorphism the Floer chain complex of a connected sum with a version of the A∞-tensor product of the modules of the summands. There is a naturally associated spectral sequence converging to the Floer groups of the connected sum whose E² page is the Tor of the Floer groups of the summands. We discuss in detail a simple example, and use this computation to show that the Pin (2)-monopole Floer homology of S³ has non-trivial Massey products.