We study the space of paths in a closed manifold M with endpoints determined by an involution \(f:M\rightarrow M\) . If the involution is fixed point free and if M is 2-connected then this path space is the universal covering space of the component of non-contractible loops of the free loop space of \(M/\mathbb {Z}_2\) . On the homology of said path space we study string topology operations which extend the Chas–Sullivan loop product and the Goresky–Hingston loop coproduct, respectively. We study the case of antipodal involution on the sphere in detail and use Morse–Bott theoretic methods to give a complete computation of the extended loop product and the extended coproduct on even-dimensional spheres. These results are then applied to prove a resonance theorem for closed geodesics on real projective space.
