Obstructions to Pin Structures on Kleinian Manifolds

We develop various topological notions on four-manifolds of Kleinian signature $(--++)$. In particular, we extend the concept of `Kleinian metric homotopy’ to non-orientable manifolds. We then derive the topological obstructions to pin-Klein cobordism, for all of the pin groups. Finally, we discuss various examples and applications which arise from this work.