Abstract

Gyrations are operations on manifolds that arise in geometric topology, where a manifold M may exhibit multiple gyrations depending on the chosen “twisting.” A natural question arises: for a given M, do all gyrations share the same diffeomorphism, homeomorphism, or homotopy type regardless of the twisting? This property is known as gyration stability. Inspired by recent work by Duan, which demonstrated that the quaternionic projective plane is not gyration stable (with respect to diffeomorphism) by invoking spin structures, we will explore gyration stability of projective planes from the homotopy theoretic perspective. Our generalization provides new results and leads to a complete description of gyration stability for the complex, quaternionic, and octonionic projective planes. This is joint work with Stephen Theriault.