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

Abstract

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

43 pages, 14 figures. Comments very welcome!

References in corpus (2)

• Manolescu Invariants of Connected Sums
• 3-manifolds Modulo Surgery Triangles

Cited by in corpus (3)

• Connected sums and involutive knot Floer homology
• $\mathrm{Pin}(2)$-Monopole Floer homology and the Rokhlin invariant
• Connected Heegaard Floer homology of sums of Seifert fibrations