A mod 2 index theorem for pin$^-$ manifolds

Understand

We establish a mod 2 index theorem for real vector bundles over 8k+2 dimensional compact pin$^-$ manifolds.

• The analytic index is the reduced $\eta$ invariant of (twisted) Dirac operators and the topological index is defined through $KO$-theory.
• Our main result extends the mod 2 index theorem of Atiyan and Singer to non-orientable manifolds.