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We study the behavior ofPin   (  2  )   ‐monopole Floer homology under connected sums. After constructing a (partially defined)    A  ∞   ‐module structure on thePin   (  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    E  2    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 thePin   (  2  )   ‐monopole Floer homology of    S  3    has non‐trivial Massey products.

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