Second-Jet Equivariant $η$ 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 $η$ 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 $η$ value. For the square family $L(\ell^2,\ell-1)$ and $L(\ell^2,2\ell-1)$, with $\ell\geq 5$ odd, we obtain a residual-circle equivariant $η$ separation: the ordinary $η$ values agree, and the first derivative of the residual $η$ 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 $η$ germ detects a distinction invisible to the ordinary $η$ invariant. The calculation uses spin-Fourier residues directly; perturbative Hessian signs serve only as motivation and are not part of the invariant.