We study the structural and dynamical consequences of introducing a distributional density profile inside a neutron star, representing a massless, topological shell located at an arbitrary radius. We incorporate this effect into the structure of neutron stars and construct an equilibrium sequence for several realistic equations of state. Radial stability is examined through the Sturm-Liouville formulation of the $\ell =0$ perturbation equation, supplemented with a jump condition and imprinting distinct features on the fundamental $f$-mode spectrum. We find strong, nonmonotonic variations in the mode frequency relative to standard no-shell models. Using first-principles scaling relations, we estimate various gravitational-wave observables such as the damping time, quality factor, luminosity, and characteristic strain. These observables are then compared with the sensitivity of Advanced LIGO and third-generation detectors such as the Einstein Telescope and Cosmic Explorer. Our results demonstrate that internal topological shells can leave potentially observable signatures in the oscillation and gravitational-wave properties of neutron stars.

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• Alternative gravity theories
• General relativity
• Gravitational waves
• Neutron stars & pulsars