Extensions of the loop product and coproduct, the space of antipodal paths and resonances of closed geodesics
Maximilian Stegemeyer
TL;DR
The work extends fundamental string topology operations to spaces P_fM determined by smooth involutions, producing an extended Chas–Sullivan product and a lifted coproduct that interrelate with closed geodesics. By exploiting Morse–Bott theory and completing manifolds, the authors give explicit computations of the extended product on odd and even spheres under antipodal involutions and derive a precise additive and multiplicative description of the homology of antipodal-path spaces. These calculations yield a resonance-type theorem for real projective spaces, tying critical values of the energy (via the length functional) to index growth of closed geodesics and establishing density results for geodesics in RP^n. The results illuminate how symmetry-induced path spaces augment string topology and provide concrete algebraic models that connect loop spaces, antipodal geometry, and geodesic resonance phenomena.
Abstract
We study the space of paths in a closed manifold $M$ with endpoints determined by an involution $f\colon M\to M$. 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.