# Any spin $M^3$, exists a natural induced $\text{Pin}^-$ structure on Poincare dual PD

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?