Second-Jet Equivariant $\eta$ Separations on Lens Spaces

This paper demonstrates that for specific families of three-dimensional lens spaces, the second-jet equivariant $\eta$ invariant distinguishes between pairs that share identical ordinary $\eta$ values and vanishing first derivatives, thereby revealing a spectral distinction invisible to the standard invariant.

The Gist

Imagine you are a detective trying to tell two identical-looking twins apart. You look at their height, weight, and shoe size, and they are exactly the same. In the world of mathematics, specifically in a field called spectral geometry, these "twins" are called Lens Spaces. They are strange, curved shapes (like a 3D donut made of a sphere) that are built using specific mathematical rules.

For a long time, mathematicians had a standard "tape measure" to check if two Lens Spaces were truly different. This tape measure is called the $\eta$ (eta) invariant. It's a single number calculated by listening to the "sound" (the spectrum) of the shape. If the numbers matched, the shapes were considered indistinguishable by this method.

The Problem: The "Blind" Tape Measure

In this paper, the author, Sanchita Sharma, discovers a pair of Lens Spaces—let's call them Space A ($L(25, 4)$) and Space B ($L(25, 9)$)—that are perfect impostors. When you use the standard tape measure (the ordinary $\eta$ invariant), they give the exact same number. They look identical.

But the author suspects they aren't actually the same. The standard tape measure is too blunt; it's like trying to tell two different songs apart by only listening to the total volume. You miss the melody.

The New Tool: The "Spin-Fourier" Microscope

To solve this, the author builds a much more sensitive tool. Instead of just measuring the total "volume" of the shape's sound, she looks at the spin of the sound waves.

Think of the shape as a spinning top. The standard measurement just counts how fast it spins. The author's new method, called Spin-Fourier residues, looks at how the top spins in different directions. It's like listening to a song not just for volume, but for the specific notes played on the violin versus the cello.

She uses a "coordinate torus action," which is a fancy way of saying she rotates the shape in two different directions independently and listens to how the sound changes in response to each specific rotation.

The Discovery: The "Second-Jet" Clue

When the author applies this new, high-resolution microscope to the two "identical" Lens Spaces, something amazing happens:

1. The First Check (Zeroth Order): The total numbers are still the same. (They are still twins).
2. The Second Check (First Derivative): She looks at how the numbers change as she slightly adjusts the rotation. Surprisingly, for both shapes, this change is zero. It's like both twins standing perfectly still when you nudge them.
3. The Third Check (Second Derivative): This is the breakthrough. She looks at the acceleration of the change—the "curvature" of the sound.
   • For Space A, the curvature is a specific number.
   • For Space B, the curvature is a different number.

The author calculates this difference precisely. For the pair $L(25, 4)$ and $L(25, 9)$, the difference in this "acceleration" is -6080.

The "Square Family" Pattern

The author doesn't just stop at one pair. She finds an infinite family of these "impostor twins." She creates a recipe using an odd number $\ell$ (like 5, 7, 9...) to generate pairs of Lens Spaces that always fool the old tape measure but always reveal their differences with her new microscope.

She proves that for every pair in this family, the standard measurement is zero, the first change is zero, but the second change is always a non-zero number. This means the shapes are mathematically distinct, even though the old tools said they were the same.

Why This Matters (According to the Paper)

The paper claims this is a second-jet separation. In simple terms, it means the author found a way to distinguish these shapes by looking at the "second derivative" of their symmetry properties.

• Old Way: "These two shapes have the same score."
• New Way: "These two shapes have the same score, and they react the same way to a gentle push, but if you push them a little harder, they react differently."

The author emphasizes that this is a purely mathematical discovery about the geometry and symmetry of these specific shapes. She explicitly states she is not creating a new medical tool or a physical device; she is refining the mathematical "language" we use to describe the universe's shapes. She uses a "perturbative" method (a theoretical nudge) only to explain why the second derivative matters, but the final proof relies on exact algebraic calculations, not approximations.

Summary

Sanchita Sharma has found a way to tell two mathematically "identical" shapes apart by listening to the subtle, hidden rhythms of their spin. She showed that while their "volume" is the same, the way their sound curves under rotation is different. This proves that these shapes are unique, even when our standard tools say they are the same.

Technical Summary

Technical Summary: Second-Jet Equivariant $\eta$ Separations on Lens Spaces

Problem Statement
Lens spaces $L(p, q)$ serve as fundamental test cases in spectral geometry due to the explicit congruence descriptions available for their spin Dirac eigenspaces. A central challenge in this field is distinguishing between lens spaces that share identical ordinary spectral invariants, such as the scalar $\eta$ invariant of Atiyah-Patodi-Singer. While the scalar $\eta$ invariant measures spectral asymmetry as a signed sum over the spectrum, it often fails to detect distinctions between lens spaces that are homotopy equivalent but not isometric. Specifically, there exist pairs of lens spaces where the ordinary scalar $\eta$ values coincide, and even the first-order derivatives of equivariant refinements vanish due to symmetry. The paper addresses the question of whether higher-order equivariant data, specifically the "second jet" of an equivariant $\eta$ germ, can detect distinctions invisible to lower-order invariants.

Methodology
The author employs a representation-theoretic approach to the spin Dirac spectrum on three-dimensional lens spaces, utilizing the framework of Bär and Gilkey. The methodology proceeds through several distinct stages:

1. Spin-Fourier Residues: Instead of merely counting the multiplicity of eigenvalues (scalar multiplicity), the paper tracks the action of the coordinate torus $T^2$ on the spinor bundles. The spin weights are described by pairs of odd integers $(a_1, a_2)$ satisfying an affine congruence $a_1 + qa_2 \equiv 0 \pmod p$. The author defines "spin-Fourier characters" $\chi(u, v)$ which retain the weight data of the lifted circle action, rather than collapsing this data to a scalar dimension.
2. Two-Variable vs. One-Variable Residues: The paper distinguishes between the full two-variable residue (tracking both coordinate weights) and the one-variable specialization (tracking only the first coordinate). It demonstrates that while one-variable residues may agree for distinct lens spaces, the full two-variable residues can differ, providing a finer invariant.
3. Residual-Circle $\eta$ Germ: The study restricts the $T^2$-equivariant $\eta$ character to a "residual circle," a one-parameter subgroup rotating the first coordinate. This yields a one-variable $\eta$ germ, $\Phi(\delta)$, defined near the identity element.
4. Perturbative vs. Invariant Calculation: The paper explicitly rejects using perturbative Hessian signs (derived from deforming the Dirac operator) as invariants, noting their dependence on the perturbation chamber. Instead, the calculation relies directly on the exact spin-Fourier residues derived from the affine congruence lattice.
5. Analytic Computation: The $\eta$ germ is expressed as a finite sum of cosecant terms involving the deck transformations. The author analyzes the Taylor expansion of this germ at $\delta = 0$.

Key Contributions and Results

• Spin-Fourier Refinement: The paper establishes that the reduced spin-Fourier residue is sensitive to the arithmetic parameter $q$ even when the ordinary scalar multiplicity and the ordinary $\eta$ invariant agree. This is demonstrated for the pair $L(25, 4)$ and $L(25, 9)$, where the scalar $\eta$ values are identical, and the same-level scalar multiplicities match, yet the reduced spin-Fourier residues differ.
• Second-Jet Separation: The primary result is the construction of a "residual-circle second-jet separation." For the pair $L(25, 4)$ and $L(25, 9)$:
  • The ordinary $\eta$ value (zeroth order) vanishes in the relative difference.
  • The first derivative of the residual $\eta$ germ vanishes due to the symmetry of the finite sum (the germ is an even function).
  • The second derivative is non-zero. In the normalized convention used, $\Phi''(0) = -6080$.
• Infinite Family: The result is generalized to an infinite family of lens spaces $L(\ell^2, \ell-1)$ and $L(\ell^2, 2\ell-1)$ for odd $\ell \ge 5$. For this family, the normalized second derivative is given by the formula:
  $$C_\ell = -\frac{\ell^2(\ell-1)^2(\ell+1)(11\ell^2 + 20\ell + 81)}{180}$$
  This coefficient is non-zero for all odd $\ell \ge 5$, proving that the residual-circle equivariant $\eta$ germ detects a distinction invisible to the ordinary $\eta$ invariant and its first derivative.
• Finite-Order Separation: The non-vanishing of the second derivative implies the existence of a finite-order element $g$ in the residual circle such that the equivariant $\eta$ values $\eta_g(D)$ for the two lens spaces are distinct.

Significance and Claims
The paper claims to provide a concrete, computable example where the ordinary scalar $\eta$ invariant and the first-order equivariant refinement fail to distinguish between lens spaces, but a second-order equivariant refinement succeeds.

• Scope of Invariance: The author is explicit that these results are "equivariant spin statements" dependent on the chosen coordinate torus action and the round metric. The paper does not claim to provide a classification theorem for lens spaces or to show that these pairs are invisible to all finite-cyclic APS $\rho$-invariants (which would require considering twists by representations of the fundamental group).
• Nature of the Invariant: The calculation is presented as a numerical separation using the "residual-circle equivariant $\eta$ germ." The author clarifies that they do not construct a Higson-Roe analytic-surgery class or a secondary K-theoretic invariant in the sense of equivariant K-homology, though the work is positioned relative to these theories.
• Motivation: The work is motivated by the need to understand the limits of scalar spectral invariants and to demonstrate the utility of retaining full spin-weight data (the "two-variable" approach) rather than discarding it for scalar shadows. The perturbative Hessian calculation is included only as motivation to explain why the second coordinate weight is relevant, but the actual invariant is derived directly from the exact residues.

In summary, the paper demonstrates that for specific families of lens spaces, the "second jet" of the equivariant $\eta$ function serves as a finer spectral invariant than the scalar $\eta$ value or its first derivative, successfully distinguishing spaces that appear identical under coarser spectral analysis.