Canonical quadratic refinements of cohomological pairings from   functorial lifts of the Wu class
  As mentioned in the comments, the proposed construction of the canonical quadratic refinement on a mapping torus contains a gap. In addition, the claim in the appendix that the spin cobordism group vanishes in dimensions 4k+3 is incorrect, although it does vanish for the case of dimension 7, relevant for physics. 
Samuel Monnier
math.AT hep-th math-ph math.MP