$\mathrm{Pin}(2)$-monopole Floer homology, higher compositions and connected sums

10 May 2016  ·  Lin Francesco ·

We study the behavior of $\mathrm{Pin}(2)$-monopole Floer homology under connected sums. After constructing a (partially defined) $\mathcal{A}_{\infty}$-module structure on the $\mathrm{Pin}(2)$-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 $\mathcal{A}_{\infty}$-tensor product of the modules of the summands... There is an associated Eilenberg-Moore spectral sequence converging to the Floer groups of the connected sum whose $E^2$ page is the $\mathrm{Tor}$ of the Floer groups of the summands. We discuss in detail a simple example, and use this computation to show that the $\mathrm{Pin}(2)$-monopole Floer homology of $S^3$ has non trivial Massey products read more

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Geometric Topology