Obstructions to Pin Structures on Kleinian Manifolds

TL;DR

This paper explores topological obstructions to pin structures on Kleinian four-manifolds, extending concepts like Kleinian metric homotopy to non-orientable cases and analyzing cobordism obstructions for all pin groups.

Contribution

It introduces new topological notions for Kleinian manifolds, extending pin structure theory and deriving obstructions to pin-Klein cobordism in non-orientable contexts.

Findings

Derived topological obstructions to pin-Klein cobordism for all pin groups

Extended Kleinian metric homotopy to non-orientable four-manifolds

Provided examples and applications of the developed theory

Abstract

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.