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"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.

There $\text{PD}(a)$ 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).

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: Are there systematic and similar statements for the induced structures on the Poincare dual manifold? Say for, inducing any one from any of the other:

  • Spin structure

  • Spin$^c$ structure

  • Spin$^h=\frac{\text{Spin} \times SU(2)}{\mathbb{Z}/2\mathbb{Z}}$ structure

  • $\text{Pin}^+$ structure

  • $\text{Pin}^-$ structure

or whatever-related structures, of

  • $\frac{\text{(S)Pin}^{\pm} \times G}{\mathbb{Z}/2\mathbb{Z}},$

where $G$ contains the same finite order-2 cyclic group ${\mathbb{Z}/2\mathbb{Z}}$ shared by the $\text{(S)Pin}^{\pm}$

in any dimension or in whatever other dimension?

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