"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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.