"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?
