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

Abstract

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 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 the Pin ( 2 ) -monopole Floer homology of S 3 has non-trivial Massey products.