Let's consider two n-dimensional closed Riemannan manifolds N and M that are cobordant to an undetermined (n+1) dimensional manifold W (That is: N and M are the boundary of W). If all considered manifolds admit spin structures, then does each class of inequivalent spin structures on N,M determine a different cobordism (and hence W)? An answer to this general question may be to long so I'll be more spedific:
An example in 1-d is easiest. Choosing two circles as our 1-dimensional N and M, We have two spin structures (the connected or unconnected double cover over each). Trivially the connected double cover gives N and M each as the boundary of a disk $D^{2}$ (see for example here ). Then our W is the disjoint union of two 2-disks.
Similarly, choosing the unconnected double cover gives N and M as the boundary of a cylinder.
(Note here, that the connected sum of these surfaces is the topological sphere, does a similar result happen in higher dimensions...?)
How much does this generalize? If I have g different spin structures on my boundaries, does each choice of spin structure correspond to a different W? If so, what is the relation between these different W, are they homeomorphic (or diffeomorphic for $n\geq3$)?
My specific case:
I am working with closed three-manifolds of genus g which each admit $2^{g}$ spin structures. How do I know which spin structures are a subset of a 4-manifold resulting from a cobordism of these manifolds? The last question is what I'm trying to answer, though an answer covering the general case would be amazing!
