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It seems that "R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows

given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural induced $\text{Pin}^-$ structure.

$\text{PD}(a)$ is a smooth, possibly non-orientable submanifold in $M^3$ representing Poincare dual to $a\in H^1(M^3,\mathbb{Z}_2)$ (it always exist in codimension 1 case).

Question 1: How do we digest this is always true?

My take is that:

  • (1) The normal bundle to the submanifold $\text{PD}(a)\equiv N^2\subset M^3$ for oriented $M^3$ can be realized as determinant line bundle $\det T{N^2}$, so that $TM^3|_{N^2}=TN^2\oplus \det TN^2$.

  • (2) For a general vector bundle $V$, there is a natural bijection between Pin$^-$- structures on $V$ and Spin-structures on $V\oplus \det V$.

Question 2: How can one show that (2) is true?

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