On the space of nondepraved (see ) real, isolated singularities, we consider the stable equivalence relation induced by smooth deformations whose asymptotic behaviour is controlled by the Palais-Smale condition. It is shown that the resulting space of equivalence classes admits a canonical semiring structure and is isomorphic to the semiring of stable homotopy classes of CW-complexes.
In an application to Hamiltonian dynamics, we relate the existence of bounded and periodic orbits on noncompact level hypersurfaces of Palais-Smale Hamiltonians with just one singularity that is nondepraved to the lack of self-duality (in the sense of E. Spanier and J. Whitehead) of the sublink of the singularity.
"Homotopical dynamics, III: Real singularities and Hamiltonian flows." Duke Math. J. 109 (1) 183 - 204, 15 July 2001. https://doi.org/10.1215/S0012-7094-01-10917-4