Second-Jet Equivariant $\hat{I}\cdot$ Separations on Lens Spaces

Abstract

Lens spaces are useful test examples in spectral geometry because their spin Dirac eigenspaces admit explicit congruence descriptions. We use these descriptions to study equivariant $\hat{I}\cdot$ invariants for three-dimensional lens spaces with the round metric and the standard coordinate-torus action, retaining the spin-Fourier character of each eigenspace rather than only the ordinary scalar $\hat{I}\cdot$ value. For the square family $L({\ell }^2,\ell -1)$ and $L({\ell }^2,2\ell -1)$, with $\ell \ge 5$ odd, we obtain a residual-circle equivariant $\hat{I}\cdot$ separation: the ordinary $\hat{I}\cdot$ values agree, and the first derivative of the residual $\hat{I}\cdot$ germ vanishes by symmetry, but the second derivative is nonzero. For $L(25,4)$ versus $L(25,9)$, the normalized second derivative is $-6080$. Thus, the residual-circle equivariant $\hat{I}\cdot$ germ detects a distinction invisible to the ordinary $\hat{I}\cdot$ invariant. The calculation uses spin-Fourier residues directly; perturbative Hessian signs serve only as motivation and are not part of the invariant.

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