M-Theory anomaly cancellation

We prove that there is no parity anomaly in M-theory in the low-energy field theory approximation. Our approach is computational. We determine generators for the 12-dimensional bordism group of pin manifolds with a w_1-twisted integer lift of w_4; these are the manifolds on which Wick-rotated M-theory exists. The anomaly cancellation comes down to computing a specific eta-invariant and cubic form on these manifolds. Of interest beyond this specific problem are our expositions of: computational techniques for eta-invariants, the algebraic theory of cubic forms, Adams spectral sequence techniques, and anomalies for spinor fields and Rarita-Schwinger fields.