PIN(2)-monopole Floer homology and the Rokhlin invariant - Lin, Francesco (Compositio mathematica, 2018)
We show that the bar version of the $\text{Pin}(2)$ -monopole Floer homology of a three-manifold $Y$ equipped with a self-conjugate spin $^{c}$ structure $\mathfrak{s}$ is determined by the triple cup product of $Y$ together with the Rokhlin invariants of the spin structures inducing $\mathfrak{s}$ . This is a manifestation of mod $2$ index theory and can be interpreted as a three-dimensional counterpart of Atiyah’s classical results regarding spin structures on Riemann surfaces.