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.

We study Fredholm properties and index formulas for Dirac operators over complete Riemannian manifolds with straight ends. An important class of examples of such manifolds are complete Riemannian manifolds with pinched negative sectional curvature and finite volume.

We prove the vanishing of the equivariant holomorphic torsion forms associated to the holomorphic de Rham complex. We also consider corresponding torsion forms associated to complexes in hypercohomology, for which we give a local formula. Rigidity of certain genera explains part of these formulas. The Lefschetz operators in Hodge theory play a prominent role in the proofs.