# Induced new structures on Poincare dual manifolds

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?