I'm looking for an example of a non-compact spin manifold $M$ and a compact subset $K\subseteq M$ such that $\partial K$ is a compact hypersurface in $M$ with $\hat{A}(\partial K)\neq 0$.


(At first I thought that $M=\mathbb{CP}^3$ and $\partial K = $ the surface defined by $x^4+y^4+z^4+w^4=0$ might be an example, but $\partial K$ isn't codimension $1$; I'm not sure what $K$ would be in this case and whether such a $K$ would be spin.)


Any suggestions for what $M,K$ and $\partial K$ could be would be appreciated!



Example of a certain partitioned manifold
Assuming that you want $K$ to be a compact manifold with boundary, no such example exists. This is because $\partial K$ is orientedly nullcobordant ($K$ is a cobordism between $\partial K$ and $\emptyset$), so its Stiefel-Whitney and Pontryagin numbers vanish. But the $\hat{A}$ genus is just a rational linear combination of Pontryagin numbers, so it also vanishes.


