In this presentation I will discuss the results of my PhD research.
Orientifolds are manifolds acted upon by a group which has a parity assigned to each of its elements.
Complex vector bundles over orientifolds carry group actions which act either linearly or anti-linearly, depending on the parity of a given group element.
I will show how to identify the obstruction to the existence of a spin-structure for the resulting K-theory using a new cohomology theory.
This enables the construction of a canonical Dirac operator with both linear and anti-linear symmetries, and a geometric K-homology theory for orientifolds.