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As mentioned in the comments, the proposed construction of the canonicalquadratic refinement on a mapping torus contains a gap. In addition, the claimin the appendix that the spin cobordism group vanishes in dimensions 4k+3 isincorrect, although it does vanish for the case of dimension 7, relevant forphysics.

Canonical quadratic refinements of cohomological pairings from functorial lifts of the Wu class