Closed geodesics and Fr{\o}yshov invariants of hyperbolic three-manifolds

Froyshov invariants are numerical invariants of rational homology three-spheres derived from gradings in monopole Floer homology. In the past few years, they have been employed to solve a wide range of problems in three and four-dimensional topology. In this paper, we look at connections with hyperbolic geometry for the class of minimal $L$-spaces. In particular, we study relations between Froyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive effective upper bounds for the Froyshov invariants of minimal hyperbolic $L$-spaces purely in terms of volume and injectivity radius. Second, we describe an algorithm to compute Froyshov invariants of minimal $L$-spaces in terms of data arising from hyperbolic geometry. As a concrete example of our method, we compute the Froyshov invariants for all spin$^c$ structures on the Seifert-Weber dodecahedral space. Along the way, we also prove several results about the eta invariants of the odd signature and Dirac operators on hyperbolic three-manifolds which might be of independent interest.

Understanding the relationship between hyperbolic geometry and Floer theoretic invariants of three-manifolds is one of the outstanding problems of low-dimensional topology. In our previous work [28], as a first step in this direction, we studied sufficient conditions for a hyperbolic rational homology sphere to be an L-space (i.e. to have simplest possible Floer homology [24]) in terms of its volume and complex length spectrum. Our approach was based on spectral geometry and its relation to hyperbolic geometry via the Selberg trace formula. It was implemented explicitly (taking as input computations from SnapPy [13]) to show that several manifolds of small volume in the Hodgson-Weeks census [20] are L-spaces.
In the present paper we focus our attention on the Frøyshov invariants of rational homology spheres. These are numerical invariants hpY, sq P Q indexed by spin c structures which are extracted from the gradings in monopole Floer homology ( [23], Chapter 39). The corresponding invariants in the context of Heegaard Floer homology are known as correction terms [42], and the identity dpY, sq "´2hpY, sq holds under the isomorphism between the theories (see [46], [15], [25] and subsequent papers). These invariants have been applied in recent years to a wide range of problems in three and four dimensional topology, see among the many [43], [44], [26]. Despite this, their computation in specific examples is still a very challenging problem, even under the assumption that Y is an L-space.
The first basic question about them in the spirit of the present paper is the following. Recall that given constants V, ε ą 0, the set of hyperbolic rational homology spheres Y for which volpY q ă V and injpY q ą ε is finite ( [5], Chapter 5), and therefore so is the set of possible Frøyshov invariants hpY, sq. Question 1. Given V, ε ą 0, can one provide an effective upper bound on |hpY, sq| for all hyperbolic rational homology spheres with volpY q ă V and injpY q ą ε?
Of course, this is a challenging question even when restricting to the smaller class of Lspaces with volpY q ă V and injpY q ą ε. Another natural question in this spirit is the following.
Question 2. Can one explicitly determine hpY, sq in terms of data coming from hyperbolic geometry (e.g. volume, injectivity radius, etc.)?
While we are not able to address these questions in the stated generality, we will answer them under the additional assumption that Y is a minimal hyperbolic L-space, i.e. a rational homology sphere pY, g hyp q equipped with a hyperbolic metric g hyp for which sufficiently small perturbations of the Seiberg-Witten equations do not admit irreducible solutions. One of the main results of [28] is that a hyperbolic rational homology sphere Y for which λ1 ą 2 (where λ1 is the first eigenvalue of the Hodge Laplacian acting on coexact 1-forms) is a minimal hyperbolic L-space; furthermore, the inequality λ1 ą 2 can be verified algorithmically in concrete examples, taking as input the length spectrum up to a certain cutoff. The first result we present, which addresses Question 1, is the following.
Theorem 0.1. Suppose Y is a minimal hyperbolic L-space with volpY q ă V and injpY q ą ε, then there exists an effectively computable constant K V,ε for which |hpY, sq| ď K V,ε for all spin c structures on Y . For example, if Y is a minimal hyperbolic L-space with volpY q ă 6.5 and injpY q ą 0.15 (e.g. any rational homology sphere in the Hodgson-Weeks census 1 with λ1 ą 2), then for every spin c structure s, the inequality |hpY, sq|ď 67658 holds. 1 The Hodgson-Weeks census [20] consists of approximatively 11 thousand closed oriented hyperbolic manifolds with volpY q ă 6.5 and injpY q ą 0.15, and most of these are rational homology spheres. It is currently not known what percentage of the such manifolds it encompasses.
In general, the dependence of K V,ε on V and ε is readily computable but does not admit a particularly pleasant closed form. In order to streamline our discussion, we take a simple approach to prove Theorem 0.1 leading to bounds which are not asymptotically optimal. More refined arguments might lead to significantly sharper estimates, especially in the case of small injectivity radius (cf. Remark 5.2).
Remark 0.1. While there are many examples of minimal L-spaces with volume ă 6.5 and injectivity radius ą 0.15, it is not known whether there are examples with arbitrarily large volume or arbitrarily small injectivity radius, cf. [28,Section 5].
The key observation behind the proof of Theorem 0.1 is the following: for minimal hyperbolic L-spaces, the Frøyshov invariant hpY, sq can be expressed in terms of the eta invariants η sign and η Dir of the odd signature operator˚d acting on coexact 1-forms and the Dirac operator D B 0 corresponding to the flat connection B t 0 on the determinant line bundle. Recall that both operators are first order, elliptic and self-adjoint, and are therefore diagonalizable in L 2 with real discrete spectrum unbounded in both directions. The eta invariant is a numerical invariant that intuitively measures the spectral asymmetry of an operator, i.e. the difference between the number of positive and negative eigenvalues [2]. Of course, in our cases of interest, both of these quantities are infinite, and the eta invariant is defined via suitable analytic continuation. While the latter was originally obtained using the heat kernel, in our setup it can also be understood in terms of closed geodesics via the Selberg trace formula for odd test functions. One should compare this with classical work of Millson [34] and Moscovici-Stanton [37] expressing the eta invariants in terms of values of suitable odd Selberg zeta functions. In particular, it is possible to provide explicit expressions for η sign and η Dir in terms of spectral and geometric data: ‚ in the case of η sign , the geometric input is the complex length spectrum we have already exploited in [28]. The main difference is that in our previous paper we only needed the trace formula for even test functions. This is because we were interested in the Hodge Laplacian ∆ acting on coexact 1-forms, and the trace formula involved spectral parameters t j with t 2 j " λj . When using even test functions, the choice of the sign t j is irrelevant, but in fact there is a natural choice (for a fixed orientation of Y ) because ∆ is the square of˚d. ‚ in the case of η Dir , the relevant new geometric data is encoded in the spin c length spectrum of pY, sq; this can be used to obtain information about the spectra of the corresponding Dirac operator via a specialization of the Selberg trace formula for the group G " tg P GL 2 pCq for which |detpgq| " 1u .
This should be thought as the spin c analogue of the group PGL 2 pCq " Isom`pH 3 q which we studied in our previous paper.
Given this, Theorem 0.1 follows by applying ideas of analytic number theory and choosing suitable compactly supported test functions. The most notable inputs are local Weyl laws [38], which allow one to effectively bound from above the number of (coexact)˚d or the number of Dirac eigenvalues in any given interval. The trace formula relates both the odd signature and Dirac eta invariants quite explicitly to the hyperbolic geometry of the underlying manifold, which allows us to prove effective upper bounds. For example, we will show that for any hyperbolic rational homology sphere Y with volume ă 6.5 and injectivity radius ą 0.15, the explicit inequalities |η sign | ď 108267 |η Dir | ď 108249 hold, where the second estimate is independent of the choice of spin c structure. Let us remark again that these estimates are not optimal even within the range of our techniques.
The injectivity radius makes its appearance in the assumptions of these results because it equals half of the length of the shortest closed geodesic. This implies that the relevant trace formula takes a particularly simple form when evaluated using test functions supported in the interval r´2¨injpY q, 2¨injpY qs, as the sum over closed geodesics vanishes.
More generally, explicit knowledge of the length spectrum up to a certain cutoff provides much more detailed information about Frøyshov invariants and can in fact be used to provide explicit computations in the spirit of Question 2. In the second part of the paper, after showcasing the main ideas behind this approach in the simple case of the Weeks manifold (whose Frøyshov invariants can be computed in a purely topological fashion [36]), we will focus on making this process explicit in the challenging case of the Seifert-Weber dodecahedral space SW. While this is one of the first examples of hyperbolic manifolds to be discovered [47], it is a complicated space to study from the point of view of three-dimensional topology; for example it took 30 years to verify Thurston's conjecture that SW is not Haken [11]. In [29] we used our techniques to show that SW is a minimal hyperbolic L-space by taking into account its large symmetry group. The next result determines its Frøyshov invariants for all the spin c structures on SW (recall that H 1 pSW; Zq " pZ{5Zq 3 q.
Theorem 0.2. Let s 0 be the unique spin structure on SW, and consider a spin c structure s " s 0`x on SW for x P H 2 pSW, Zq. Then the Frøyshov invariant hpSW, sq is computed as in the following table, according to the value of the linking form lkpx, xq P Q{Z.
An important observation here is that the fractional parts of Frøyshov invariants of Y admit partial interpretations in terms of the linking form of Y and the topology of manifolds bounding Y ; given the extra topological input, the problem boils down to computing eta invariants up to a certain small (but reasonable) error. Towards this end, the main limitation is that we can only access a limited amount of the length spectrum of SW. To prove Theorem 0.2, it is most convenient to use dilated Gaussians as test functions, because both the function and its Fourier transform (which is again a Gaussian) are rapidly decaying; as Gaussians are not compactly supported, we need to truncate certain infinite sums over closed geodesics that arise when estimating the η sign and η Dir via the odd trace formula for coexact 1-forms and spinors. The main step in the proof is then to estimate the error introduced in this procedure. This can be done by deriving effective bounds on the number of closed geodesics of a given length, in the spirit of the prime geodesic theorem with error terms ( [10], Section 9.6).
Remark 0.3. The approach we implement for SW can be in principle carried out for any minimal hyperbolic L-space for which the linking form is known; the fundamental limitation comes from computing length spectra. In fact, the same ideas can also be exploited to obtain closed formulas for the Frøyshov invariants of minimal hyperbolic L-spaces with finitely many terms with an effective (but impractical) upper bound on the number of terms. We will not pursue such a closed formula in the present paper.
Let us remark that explicit computations for the eta invariant of the odd signature operator η sign for hyperbolic three-manifolds have been implemented in Snap [16], and are based on a Dehn filling approach [40]. The key insight is that for a fixed oriented compact 4-manifold X bounding Y , the general Atiyah-Patodi-Singer index theorem [2] relates η sign to the kernel of the odd signature operator on Y and the signature of X, both of which are topological invariants.
However, while the APS index theorem also holds for the spin c Dirac operator, it is well known that the dimension of its kernel on Y is not a topological invariant [19]; this fact makes the computation of η Dir much more subtle than its odd signature operator counterpart. In fact, the computations of Frøyshov invariants carried out in this paper for some explicit minimal hyperbolic L-spaces provide as a byproduct explicit examples of hyperbolic threemanifolds for which one can compute Dirac eta invariants to high accuracy (which are not zero for obvious geometric reasons, e.g. the existence of an orientation-reversing isometry), and to the best of our knowledge these are the first such examples. For example, our methods will show that for the unique spin structure on the Weeks manifold, η Dir " 0.989992 . . . This relies on the fact that the Weeks manifold is a minimal hyperbolic L-space, together with the computation of η sign given in [16]; in particular, the value is as precise as the computations provided by Snap.
More generally, the odd trace formula allows one to obtain bounds on η Dir provided one can access the spin c length spectrum of pY, sq; in turn, we will describe an algorithm to compute spin c length spectra taking as input information computed using SnapPy. In particular, our method could in principle be applied to compute the invariants η Dir for any hyperbolic three-manifold, even though at a practical level it might be infeasible to obtain a decent approximation in a reasonable time.
Note for the reader. The paper is structured so that the various trace formulas (see Section 3 for the statements) can be treated as black boxes, and all subsequent sections are written in a way that is hopefully self-contained. In particular, in Section 4 we only use some complex analysis to provide an explicit formula for the eta invariants in terms of eigenvalues and complex (spin c ) lengths. Given this, the remainder of the paper only uses basic facts about Fourier transforms, and we will provide motivation and context for the tools from analytic number theory which we employ. Detailed proofs of the various trace formulas which we use can be found in the appendices; our discussion there assumes the reader to be familiar with the proof of the even trace formula for coexact 1-forms in our previous work [28,Appendix B].
Plan of the paper. Sections 1, 2 and 3 provide background about the main protagonists of the paper: Frøyshov invariants, spin c length spectra, and trace formulas (both even and odd). In Section 4 we use the odd trace formulas to provide an explicit expression for the eta invariants of the odd signature and Dirac operators; this will be the main tool for the present paper. In particular we prove Theorem 0.1 in Sections 5 and 6 by providing explicit bounds on the terms appearing in the sum. In Section 7, we show how our analytical expressions for eta invariants can be used to perform explicit computations on the Weeks manifold, the simplest minimal hyperbolic L-space. This example is propaedeutic for the significantly more challenging case of the Seifert-Weber space, which we discuss in detail in Sections 8, 9 and 10.

Background on Frøyshov and eta invariants
In this section we review some background topics that will be central for the purposes of the paper.

Ñ¨¨¨.
These are read respectively HM-to, HM-from and HM-bar, and we collectively refer to them as monopole Floer homology groups. Such invariants decompose along spin c structures on Y ; for example we have y HM˚pY q " à sPSpin c pY q y HM˚pY, sq.
The reduced Floer homology group HM˚pY, sq is defined to be the kernel of the map p˚.
In this paper, we will be only interested in the case of rational homology spheres. In this situation the Floer homology groups for a fixed spin c struture admit an absolute grading by a Z-coset in Q, and the action of U has degree´2. Furthermore, we have that as graded modules (up to an overall shift), y HM˚pY, sq vanishes in degrees high enough, and p˚is an isomorphism in degrees low enough.
A rational homology sphere Y is called an L-space if HM˚pY, sq " 0 for all spin c structures. Given a spin c rational homology sphere pY, sq, denote the minimum degree of a non-zero element in i˚`HM˚pY, sq˘Ă y HM˚pY, sq by´2hpY, sq. The quantity hpY, sq is then called the Frøyshov invariant of pY, sq.

Frøyshov invariants in terms of eta invariants.
Recall [2,Theorem 4.14] that for an oriented three-manifold the odd signature operator B acts on even forms Ω even as p´1q p p˚d´d˚q on 2p-forms.
We identify 2-forms and 1-forms using˚, so that the operator iŝ˚d d d˚0ȧ cting on Ω 1 ' Ω 0 . This is a first order elliptic self-adjoint operator, and is diagonalizable in L 2 with real discrete spectrum unbounded in both directions. Its eta function is defined to be this sum defines a holomorphic function for Repsq large. One of the key results of [2] is that it admits an meromorphic continuation to the entire complex plane, with a regular value at s " 0. In Section 4, we will quickly review the original proof of this (via the heat kernel) and then provide an alternative interpretation via the trace formula. Intuitively, η sign " η sign p0q measures the spectral asymmetry of the operator. Similarly, the same procedure works for the Dirac operator D B 0 (and its perturbations), leading to η Dir .
Remark 1.1. For our purposes it is convenient to notice (cf. [2,Proposition 4.20]) that the odd signature operator (assuming for simplicity b 1 pY q " 0) under the Hodge decomposition Ω 1 " dΩ 0 ' d˚Ω 2 can be written as¨˚d 0 0 0 0 d 0 d˚0‚ acting on d˚Ω 2 ' dΩ 0 ' Ω 0 . Now, the block 0 d d˚0ḣ as symmetric spectrum because d and d˚are adjoints; in particular we have for Repsq large enough where the sum runs only on the eigenvalues tt j u of˚d on coexact 1-forms (notice that all the t j are non-zero). In particular, η sign coincides with the spectral asymmetry of the action of d on coexact 1-forms.
Remark 1.2. We have p˚dq 2 " ∆ when acting on coexact 1-forms, and therefore the squares of the parameters t j are exactly the eigenvalues λj of ∆ we studied in [28]. The crucial extra information for the purposes of the present paper is the sign of these parameters.
The relation between eta invariants and the Frøyshov invariant is the following. Recall that a minimal L-space is a rational homology sphere admitting a metric for which small perturbations of the Seiberg-Witten equations do not have irreducible solutions. Classical examples of minimal L-spaces are rational homology spheres admitting metrics with positive scalar curvature. In [28], we showed that a hyperbolic rational homology sphere for which the first eigenvalue of the Hodge Laplacian coexact 1-forms λ1 satisfies λ1 ą 2 is a minimal L-space (using the hyperbolic metric); we furthermore provided several examples of such spaces.
While Proposition 1.1 is well-known to experts, we will dedicate the rest of the section to its proof; our discussion will assume some familiarity with the content of [23], and will not be needed later in the paper except in Section 10 where we will briefly use the explicit form of the absolute grading (Equation (3) below). The main idea behind the proof is the following. Under the assumption that there are no irreducible solutions, after a small perturbation the Floer chain complex has generator corresponding to the positive eigenspaces of (a small perturbation of) the Dirac operator D B 0 . From this description it readily follows that they are L-spaces [23,Chapter 22.7], and that´2hpY, sq is the absolute grading of the critical point corresponding to the first positive eigenvalue; the goal is then to express this absolute grading in terms of eta invariants using the APS index theorem.
1.3. Proof of Proposition 1.1. We begin by recalling from [23,Chapter 28.3] how absolute gradings in monopole Floer homology are defined for torsion spin c structures. As we only consider rational homology spheres, the spin c structures in our context are automatically torsion. Consider a cobordism pW, s W q from S 3 to pY, sq; equip S 3 with a round metric and a small admissible perturbation, and denote by ra 0 s the first stable critical point of S 3 , corresponding to the first positive eigenvalue of the Dirac operator. Consider on W a metric which is a product near the boundary, and denote by W˚the manifold obtained by attaching cylindrical ends. We then define for a critical point ras of Y the rational number (2) gr Q prasq "´grpra 0 s, W˚, s W , rasq`c 1 ps W q 2´2 χpW q´3σpW q 4 where: ‚ grpra 0 s, W˚, s W , rasq is the expected dimension of the moduli space of solutions of Seiberg-Witten equations in the spin c structure s W that converge to ra 0 s and ras. Concretely, this is the index of the linearized equations, after gauge fixing. ‚ c 1 ps W q 2 P Q is the self-intersection of the class c 1 ps W q P H 2 pW ; Zq. Recall that this is defined as pc YcqrW, BW s wherec P H 2 pW, BW, Qq is any class whose image in H 2 pW, Qq is the same as the image c 1 ps W q under the change of coefficient map H 2 pW, Zq Ñ H 2 pW, Qq. For our purposes, it will be convenient to work with a closed manifold X with boundary Y over which s extends instead; this can be obtained from W by gluing in a ball D 4 to fill the S 3 boundary component. In this case the formula (3) gr Q prbsq "´grpX˚, s X , rbsq`c 1 ps X q 2´2 χpXq´3σpXq´2 4 P Q holds, where grpX˚, s X , rbsq is again defined as the expected dimension of the relevant moduli space. This readily follows via the excision principle for the index from the definition in formula (2) and the computation for D 4 in [23,Chapter 27.4].
Let us point out the following observation.
Lemma 1.2. Suppose pY, gq is a minimal L-space. Then for each spin c structure s, the Dirac operator D B 0 corresponding to the flat connection B t 0 has no kernel. Remark 1.3. As a consequence, the minimal hyperbolic L-spaces we exhibited in [28] (e.g. the Weeks manifold) do not admit harmonic spinors. These seem to be the first known examples of hyperbolic three-manifolds having no harmonic spinors; more examples can be found using the trace formula techniques we discuss later in this paper.
Proof. Consider the small perturbation of the Chern-Simons-Dirac functional for which the corresponding perturbed Dirac operator at pB 0 , 0q is D B 0`δ . For small values of˘δ, this operator has no kernel and the Seiberg-Witten equations still have no irreducible solutions (because Y is a minimal L-space). By adding additional small pertubations, we can assure that the spectra of these two operators are simple, and they still do not have kernel, so we obtain transversality in the sense of [23,Chapter 12]. In particular, they both determine chain complexes computing the Floer homology, hence both the two first stable critical points have the same absolute grading´2hpY, sq. This implies that the moduli space of solutions to the perturbed equations connecting them on a product cobordism RˆY (for which the induced map is an isomorphism) is zero dimensional, hence the spectral flow of the corresponding linearized operator at the reducible is also zero. After performing a small homotopy (cf. [23,Chapter 14]), this implies that the spectral flow for a family of operators of the form D B 0`f ptq, where f ptq is a monotone function with f ptq "˘δ for t ą 1 and t ă´1 respectively, equals zero. In particular, D B 0 has no kernel.
In this argument, one can avoid adding extra small perturbations to make the spectra simple when working with Morse-Bott singularities instead [28]. To streamline the argument below, we will assume that the first positive eigenspace of D B 0 is simple, and the general case can be dealt with by either adding a small perturbation or by working in a Morse-Bott setting. Consider first reducible critical point rb 0 s, which lies over the reducible solution rB 0 , 0s. Under our assumptions, its absolute grading is exactly´2hpY, sq. Taking A to be a spin c connection of s X restricting to B 0 in a neighborhood of the boundary, linearizing the equations at pA, 0q we have The Atiyah-Patodi-Singer for the odd signature operator says in our case where p 1 is the first Pontryagin form of the metric, and η sign is the eta invariant of the odd signature operator. Let us point out that the different sign comes from our convention for boundary orientations from that of [2]: we think of X as a 'filling' of Y , rather than a 'cap', and this in turns implies that our boundary conditions involve the negative spectral projection [23,Chapter 17]. For the Dirac operator, we obtain ind C L 2 pXq pDÀq " here c 1 pAq is the Chern form of A, and the term involving the dimension of the kernel of D B 0 is not present because of Lemma 1.2. Putting everything together, we see that gr Q prb 0 sq " 1 4 η sign`ηDir , and the result follows.
2. Torsion spin c structures and spin c length spectra Recall that a closed geodesic γ in an oriented hyperbolic three-manifold admits complex length C pγq :" pγq`i¨holpγq P R ą0`i R{2πZ. In this section we discuss a refinement of this notion when the manifold is equipped with a torsion spin c structure; this refinement of complex length appears on the geometric side of the trace formula for spinors. We also discuss how the refinement can be computed in explicit examples, taking as input SnapPy's Dirichlet domain and complex length spectrum computations.
2.1. Torsion spin c structures and G. Let us begin by discussing with the simpler case of a genuine spin structure s 0 on Y . The choice of a spin structure s 0 allows us to lift the holonomy of a closed geodesic γ : S 1 Ñ Y from an element in R{2πZ to an element in R{4πZ. From a topological perspective, this is because, using the trivial spin structure on the tangent bundle of γ, s 0 this induces a spin structure on the normal bundle of γ; and a spin structure on a rank 2 real vector is simply a trivialization well defined up to adding an even number of twists (see [21,Chapter IV]). In particular, it makes sense to talk about holpγq{2 as an element in R{2πZ.
Remark 2.1. Recall that (confusingly) the trivial spin structure on S 1 corresponds to the non-trivial double cover of S 1 ; with this convention, we have holpγ n q{2 " n¨pholpγq{2q in R{2πZ. This does not hold if we choose the Lie spin structure.
From the point of view of hyperbolic geometry, thinking of π 1 pY q Ă Isom`pH 3 q -PSL 2 pCq, a spin structure is a lift where the vertical map is the quotient by the subgroup t˘1u. Notice that all orientable 3manifolds are spin because their second Stiefel-Whitney class w 2 vanishes [35, Chapter 11], so such a lift always exists. Any two such lifts differ by a homomorphism π 1 pY q Ñ t˘1u, i.e. an element in H 1 pY, Z{2q. From this viewpoint, recall that complex length of a hyperbolic element γ is given by C pγq " 2 log λ γ where λ γ is the root of x 2´p trγqx`1 " 0 with |λ γ | ą 1. Of course, for an element γ P PSL 2 pCq the trace is well-defined only up to sign, and we see that the two choices of the trace correspond to˘λ γ ; this implies that the argument of λ 2 γ , which is holpγq, is well defined modulo 2π. It is then clear that a spin structure, which provides a well-defined trace, also fixes a choice between˘λ γ ; the argument of this choice, which is holpγq{2, is well defined modulo 2π. Very concretely, this is implies that the liftγ P SL 2 pCq is conjugate tõ The case of general torsion spin c structures is analogous, where we use instead the group There is a natural map π : G Ñ PSL 2 pCq obtained by expressing G " pSL 2 pCqˆU 1 q {t˘1u, where U 1 Ă G is the subgroup of multiples of the identity, and projecting onto the first factor. A torsion spin c structure, together with a flat spin c connection, is then a homomorphism ϕ : π 1 pY q Ñ G for which π˝ϕ : π 1 pY q Ñ PSL 2 pCq is the inclusion map. We see that two spin c structures then differ by a homomorphism π 1 pY q Ñ U 1 , which can be thought of as an element in H 1 pY, R{Zq. The Bockstein long exact sequence for the coefficients 0 Ñ Z Ñ R Ñ R{Z Ñ 0 reads in this case The latter allows us to interpret topological classes of torsion spin c structure as an affine space over the torsion of H 2 pY ; Zq.
Throughout the paper, we will study rational homology spheres, for which all spin c structures are torsion. In this case, we have identifications We will further fix a base spin structure s 0 and accordingly identify spin c structures with H 1 pY, Q{Zq. For simplicity, ϕ will interchangeably refer to both the lift and the difference homomorphism. In this case, the element γ has lift conjugate tõ e iϕpγq e nd the spin c length spectrum keeps track of the value ϕpγq. We will refer to the latter as the twisting character.

2.2.
Computations of spin c lifts. For eventual use of the trace formula for spinors on G, we need to explicitly compute the torsion spin c data, namely holpγq 2 and φpγq for representatives of all conjugacy classes in Γ up to some specified real length. We refer to this data as the spin c length spectrum of pY, sq. We discuss a concrete algorithm to compute such information, taking as input data provided by SnapPy.
2.2.1. Input for spin c length spectrum: SnapPy Dirichlet domain and complex length spectrum. In order to compute the spin c length spectrum up to some real length threshold R, we take as input the following objects (both pre-computable in SnapPy): ‚ A Dirichlet domain D for Γ " π 1 pY q centered at o " p1, 0, 0, 0q in the hyperboloid model of H 3 , i.e. the upper sheet of " pt, x, y, zq such that Qpt, x, y, zq "´1 where Qpt, x, y, zq "´t 2`x2`y2`z2 . In particular, this includes a list of all elements γ of the identity component of the orthogonal group SOpQq 0 for which some domain within the bisector of o and γo is a face F Ă D. For hyperbolic 3-manifolds Y, such elements γ come in inverse pairs: for the above element γ, γ´1F is the face of D opposite F. The group Γ is generated by the face-pairing elements (see §2.2.4). ‚ For each conjugacy class C in Γ with pCq ď R, a matrix g C in SOpQq 0 , which is the image in SOpQq 0 of some element of Γ representing C. One can extract the complex length of C from g C , but of course it contains more information.
Under this identification, the quadratic form Q equals the determinant of the above matrix. The group PSL 2 pCq acts on iu 2 by g¨X " gXg˚, preserving the determinant. The resulting homomorphism is in fact an isomorphism. To transform Γ to a corresponding subgroup of PSL 2 pCq amounts to inverting the above isomorphism. In turn, this amounts to solving the following linear algebra problem: Given f P Endpiu 2 q preserving Q, find g P GL 2 pCq satisfying (4) Xg˚"`detpgq¨g´1˘f pXq for all X P iu 2 .
The entries of detpgq¨g´1 are linear in the entries of g, so (4) defines a linear algebra problem (over R), which is readily solved. Rescaling the solution g to (4) as needed, we solve (4) with some matrix g P PSL 2 pCq.

2.2.3.
Reduction theory via Dirichlet domains for conjugacy classes C Ă Γ. We discuss how to express a given element γ P Γ in terms of the face-pairing elements γ i of D.
Lemma 2.1. Suppose x P H 3 zD. There is some face pairing element γ for which dpγ´1¨x, oq ă dpx, oq.
In particular, if x R D, then dpx, oq ą dpx, γ i¨o q " dpγ´1 i¨x , oq for some face-pairing element γ i .
This yields the following Dirichlet domain reduction algorithm for Γ: ‚ Initialize x " γ¨o. Let L " rs be an empty list. Note: if γ is not the identity element, then x R D. ‚ While x is not in D: find a face pairing element γ for which dpγ´1x, oq ă dpx, oq; this is always possible by Lemma 2.1. Then: -Append γ to the right end of L.
-Replace x by γ´1x. ‚ Once x P D, output δ " ś gPL g, product taken in left-to-right order. Lemma 2.2. The Dirichlet domain reduction algorithm for Γ terminates. Its output equals γ.
Proof. Let x " γ¨o. If the while loop did not terminate, we would find an infinite sequence of face pairing elements γ 1 , γ 2 , . . . for which x, γ´1 1¨x , γ´1 2¨γ´1 1¨x , . . . all do not lie in D and for which dpx, oq ą dpγ´1 1¨x , oq ą dpγ´1 2¨x , oq ą¨¨B ut the latter is not possible since the action of Γ on H 3 is properly discontinuous. Suppose now L " rγ 1 ,¨¨¨, γ n s at the point of the algorithm when x P D. That means that γ´1 n γ´1 n´1¨¨¨γ´1 1¨p γ¨oq P D. Since the action of Γ on H 3 is fixed-point free, it follows that γ " γ 1¨¨¨γn .

2.2.4.
Computing one spin structure. As mentioned above, it is convenient to fix a base genuine spin structure s 0 (so that all the others spin c structures are obtained by twisting it via a character). To compute the corresponding liftΓ Ă SL 2 pCq, we use the presentation of Γ Ă PSL 2 pCq afforded to us by the Dirichlet domain D. Let Γ abs denote the (abstract) group with a generator rγs for each face pairing element γ and satisfying the following relations: ‚ (opposite face relations) rγs¨rγ´1s " 1 for all face pairing elements γ ‚ (edge cycle relations) The edges of D are partitioned into edge cycles. This is an ordered sequence of edges e 1 , e 2 , . . . , e n for which γ 1¨e1 " e 2 , γ 2¨e2 " e 3 , . . . , γ n¨en " e 1 for some γ 1 , γ 2 . . . , γ n . Every edge cycle yields a corresponding relation: rγ n s¨¨¨rγ 1 s " 1.
The opposite face and edge cycle relations hold for the corresponding elements of Γ. The assignment ι : Γ abs Ñ Γ Ă PSL 2 pCq rγs Þ Ñ γ (5) thus extends to a well-defined homomorphism. According to the Poincaré polyhedron theorem [30], ι is an isomorphism.
For every face-pairing element γ P Γ Ă PSL 2 pCq, choose arbitrarily a lift r γ P SL 2 pCq. Because ι is well-defined, all of the defining relations have an associated sign. For example, if rγ n s¨¨¨rγ 1 s " 1 is an edge cycle relation, then where rγns¨¨¨rγ 1 s "˘1. Let rγs "˘1 be unknowns we will solve for. Then γ Þ Ñ rγs¨r γ extends to a well-defined lift of Γ to r Γ Ă SL 2 pCq iff the opposite face and edge cycle relations are satisfied. This immediately reduces to the following linear algebra problem (over Z{2Z): # rγs rγ´1s " rγs¨rγ´1s opposite face relations rγns¨¨¨ rγ 1 s " rγns¨¨¨rγ 1 s edge cycle relations. We know abstractly that some solution must exist (because Y is spin), and we readily solve for one.
Remark 2.2. The collection of lifts r Γ is a torsor for H 1 pΓzH 3 , Z{2q. No preferred lift exists in general. We note, however, that SW, the protagonist of the second part of our paper, satisfies H 1 pSW, Z{2q " 0, so π 1 pSWq admits a unique lift to Č π 1 pSWq Ă SL 2 pCq.

2.2.5.
Computing all homomorphisms Γ Ñ Q{Z. From the previous subsection, Γ abs gives a presentation for Γ. Let e 1 ,¨¨¨, e m denote the images of the face-pairing elements γ 1 , . . . , γ m in the abelianization Γ ab . The abelian group Γ ab can be presented as Zxe 1 , e 2 , . . . , e m y{R, where R are the abelian versions of the opposite face and edge cycle relations from Subsection 2.2.4. We assume that b 1 pY q " 0, so that Γ ab " H 1 pY, Zq is a finite group. Using the Smith normal form, we compute a basis b 1 , . . . , b m of Zxe 1 , e 2 , . . . , e m y for which a basis for the span of R is given Homomorphisms from Γ ab to Q{Z are uniquely determined by the images y :" py 1 , . . . , y m q of b 1 , . . . , b m ; we define φ y as the one sending If A " pb 1 |¨¨¨|b m q is the matrix with jth column b j , then yA´1 " pφ y pe 1 q, . . . , φ y pe m qq.

2.2.6.
Indexing all lifts of Γ to r Γ Ă G. We index the lifts of Γ Ă PSL 2 pCq to r Γ Ă G as follows: ‚ Compute one spin lift s 0 : Γ Ñ Γ spin Ă SL 2 pCq, by the procedure described in §2.2.4. ‚ A general lift of Γ to G is of the form γ Þ Ñ φpγq¨s 0 pγq for some twisting character * parametrizes all lifts of Γ to G.

2.2.7.
Computing the spin c length spectrum. Suppose we have computed one lift s 0 : Γ Ñ SL 2 pCq specified by the images of the face-pairing generators for D; SnapPy provides the images in SOpQq 0 for all of the face-pairing generators, so we can apply the inverse isomorphism from §2.2.2 to each face-pairing generator followed by the procedure described in §2.2.4 to compute a lift s 0 . Suppose also that we have computed a homomorphism φ : Γ Ñ Q{Z, specified by its values on the face-pairing generators, e.g. the homomorphisms φ y from §2.2.5.
For every Γ-conjugacy class of translation length at most R, SnapPy specifies the image in SOpQq 0 of some representative element γ of its conjugacy class. Notice that γ lies in the group generated by the face-pairing elements of the Dirichlet domain D. Applying the Dirichlet domain reduction algorithm from §2.2.3, we can express γ " γ 1¨¨¨γk for some face-pairing elements γ 1 , . . . , γ k for D. Having computed s 0 on the face-pairing generators, s 0 pγq " s 0 pγ 1 q¨¨¨s 0 pγ k q is some explicit element of SL 2 pCq. We readily compute its SL 2 pCqconjugacy class: e´u {2 e´i θ˙, and we have that u " lpγq and θ " holpγq{2 P R{p2πZq.
Finally, since the values of the twisting character φ are known on the face-pairing generators, we readily compute φpγq " φpγ 1 q`¨¨¨`φpγ k q ": ϕpγq. In the lift r Γ Ă G of Γ corresponding to s 0 and φ, we have Following this procedure for every conjugacy class C of length at most R computes the spin c length spectrum for the lift r Γ Ă G of Γ corresponding to the pair s 0 : Γ Ñ SL 2 pCq and φ : Γ Ñ Q{Z.

Trace formulas for functions, forms and spinors
While our previous work [28] only used the trace formula to sample coexact 1-form eigenvalues with even test functions, the present paper requires that we extend our toolkit to sample the eigenvalue spectrum for functions and spinors (the latter using both even and odd test functions). In this section, we collect statements of the trace formula specialized to the latter three contexts. For the purposes of our work, the various statements can be treated as a black box, and their detailed proofs can be found in the appendices to the paper. For simplicity, in the statements we will restrict to smooth compactly supported functions, but we will ultimately need to apply the trace formula using more general test functions; this is made precise in Subsection 3.5.

Notation and conventions.
‚ Throughout the paper, we will use the convention for Fourier tranforms ‚ For a closed geodesic γ, we denote by γ 0 a prime geodesic of which γ is a multiple of. ‚ When dealing with the odd trace formula, it will be important to have a clear orientation convention, as for example the spectrum of˚d on Y andȲ are opposite to each other. We use the identification H 3 " PSL 2 {PSU 2 (obtained by thinking of the upper half plane model), for which the tangent space at p0, 1q is isup2q. We declare the (physicists') Pauli matrices o be a positively oriented basis.

3.2.
Trace formula for functions. We begin with the classical trace formula for functions.
Here, to each eigenvalue 0 " λ 0 ă λ 1 ď λ 2 ď . . . we associate the parameter r n P R Y ir0, 1s for which λ n " 1`r 2 n (in particular, λ 0 " i). Non-zero eigenvalues less than 1 (i.e. such that r n P ip0, 1q ) are referred to as small ; the value 1 is important because it is the bottom of the L 2 -spectrum of the Laplacian on H 3 (see [14]).
c pRq an even test function, the identity This formula is well-known, and essentially follows from the computation in [28,Appendix B]. It can also be proved by direct integration via the Abel transform and integral kernels, see for example [14] (beware of an erroneous extra factor of 2).

3.3.
Trace formula for coexact 1-forms. In the case of coexact 1-forms, each eigenvalue are the eigenvalues of˚d. In our previous paper, we were only concerned with the absolute values |t j | of the parameters and so even test functions sufficed for our purposes. In the present paper, however, the signs of the parameters t j are crucial. Below, we state a variant of the trace formula which samples the coexact 1-form eigenvalue spectrum using odd test functions and is thus sensitive to these signs.

Theorem 3.2.
For H P C 8 c pRq an even test function, the identity holds. For K P C 8 c pRq an odd test function, the identity 1 2 holds.
For the last identity, recall that if K is odd then its Fourier transform is purely imaginary.
3.4. Trace formula for spinors. As in Section 2, we fix a base spin structure s 0 and consider its twist by a character ϕ. The corresponding Dirac operator D B 0 has discrete spectrum¨¨ď unbounded in both directions. Also, recall that we introduced the notationγ for the lift of γ corresponding to the base spin structure s 0 .

Theorem 3.3.
For H P C 8 c pRq an even test function, the identity 1 2 holds. For K P C 8 c pRq an odd test function, the identity 1 2 holds.

3.5.
Allowing less regular functions. While for simplicity we stated the formulas only for smooth, compactly supported functions, the same conclusions hold with less regularity. For example, in our previous work [28, Appendix C] we showed that the trace formula for coexact 1-forms holds for functions of the form`1 r´a,as˘˚4 , the fourth convolution power of the indicator function of the interval r´a, as. Rather than providing general statements, let us point out two specific instances that will be used in this work: (1) the even trace formulas hold for Hpxq "`1 r´a,as˘˚k , with k ě 6 and the odd trace formulas hold for K " H 1 . This follows directly from the results in [28, Appendix C]. (2) the even trace formulas hold for a Gaussian Hpxq " e´x 2 {2c , and the odd trace formulas hold for K " H 1 . This is proved in Appendix D.
In the proof of Theorem 0.1 we will only use functions of the first type, but our proof of Theorem 0.2 uses Gaussians.

Analytic continuation of the eta function via the odd trace formula
The classical approach to the analytic continuation of the eta function (1) involves the Mellin transform and the asymptotic expansion of the trace of the heat kernel; we start by quickly reviewing the fundamental aspects of this for the reader's convenience. For our purposes, we will discuss a different interpretation using the odd trace formulas described in the previous section.

Analytic continuation via the Mellin transform.
For the reader's convenience, we recall basic facts about the Mellin transform and its relevance for the definition of the η invariant. Our account loosely follows [14,Section 3.4], suitably adapted to the simpler situation we are dealing with. Given a continuous function f : R ą0 Ñ R, we define its Mellin transform to be This is essentially the Fourier transform for the multiplicative group R ą0 equipped with the Haar measure dt{t. Assuming that f ptq " Opt´aq for t Ñ 0, and f ptq " Opt´nq for t Ñ 8 for all n, we have that the integral absolutely converges on the half-plane Repsq ą a, and that M f is a holomorphic function there. Under additional assumptions, M f admits an explicit analytic continuation to the whole plane. For our purposes, suppose for example that f admits an asymptotic expansion where ti k u is a strictly increasing sequence of real numbers with lim i k "`8. Then M f extends to a meromorphic function on C with a simple pole at´i k with residue c k for all k, and holomorphic everywhere else. To see this, choose L ą 0 and let us write Our assumption on the behavior of f at infinity implies that M`is an entire function. Regarding the first integral, the asymptotic expansion (6) implies that for a given N we can write Integrating, for Repsq ą´i N we can therefore write M´psq " This expression provides a meromorphic continuation of M´to the half-plane Repsq ą´i N . This continuation has simple poles at´i 0 ,¨¨¨´i N´1 with residues c 0 , . . . c N´1 and is holomorphic everywhere else in the half-plane. Because our choice of N was arbitrary, this proves the claim. The above discussion readily generalizes to the case in which f ptq " Opt´nq for some fixed n when t Ñ 8. In this case, the analytic continuation (obtained again by breaking the integral M f psq " M´psq`M`psq) defines a meromorphic continuation on the half-plane Repsq ă n.

4.2.
The heat kernel. We briefly recall the classical analytic continuation of the eta function of the odd signature operator using the asymptotic expansion of the heat kernel, see [2] for details. The key observation to relate this to the previous discussion is the identity where Γpsq " ş 8 0 e´tt s dt t is the classical Gamma function.
In the simpler case of the Riemann zeta function, we have for Repsq ą 1 (which ensures absolute convergence) the identity ζpsq " Recall that Γpsq has poles at s " 0,´1,´2, . . . with residue p´1q k {k at s "´k. Given the asymptotic expansion for t Ñ 0 in terms of Bernoulli numbers and 1{pe t´1 q " Opt´nq for all n when t Ñ 8, we recover the classical fact that ζpsq admits a meromorphic extension with only one pole at s " 1.
Let us now focus on the case of the eta function of the odd signature operator (the case of the Dirac operator is analogous). In [2], the authors study the quantity 2 is the complementary error function. As 0 is not an eigenvalue by assumption, Weyl's law (cf. Section 5) implies that Kptq Ñ 0 exponentially fast for t Ñ 8. The derivative of Kptq is computed to be Using p8q and integrating by parts we get the formula The key point is that Kptq admits an asymptotic expansion of the form (10) Kptq " ÿ kě´n a k t k{2 for t Ñ 0, and therefore its Mellin transform has at worst a simple pole at 0. In p9q, this pole is canceled by the term s in the numerator, and we see therefore that η sign psq is holomorphic near 0. We conclude by mentioning that the asymptotic expansion p10q follows by interpreting K as the trace of the heat kernel on R ě0ˆY with the (now-called) APS boundary conditions.

4.3.
Analytic continuation via the trace formula. While the analytic continuation via the heat kernel we have just discussed holds in general, for the specific case of hyperbolic three-manifolds we can take a different route using the odd trace formula. Let us discuss first in detail the case of the eta invariant for the odd signature operator (the case of the Dirac operator is essentially the same). Recall that in this case we only need to consider the spectral asymmetry of˚d acting on coexact 1-forms (1). Given an even test function H (in our case H will either be a Gaussian or compactly supported, as in Subsection 3.5), we consider the trace formula applied to the odd test function H 1 . Recalling that with our convention we obtain the identity For a fixed even test function G, and given T ą 0, we apply this to the function Hpxq " Gpx{T q, for which p Hptq " T p GpT tq, and obtain the family of identities We will denote either side of the identity by G T (and refer to them as spectral and geometric respectively). Now, for any real number a ‰ 0 we have the identity where we used that p G is even in the second equality and performed a change of variables in the last equality above. Therefore, for Repsq large enough (so that the sums converge absolutely), we have We can recognize at the numerator and denominator the Mellin transforms of G T and T p GpT q respectively. We have the following.
‚ G is the k-th convolution power of an indicator function 1 2a 1 r´a,as , for some even n ě 6. Then the identity η sign psq " holds in an open domain in C containing s " 0, where we interpret the right hand side using the Mellin transform.
Of course, the lemma is valid in much greater generality, but for simplicity we have restricted to the class of functions that will be used in the rest of the paper.
Proof. Let us start with the case of a Gaussian. We claim that the quantity G T is rapidly decaying for both T Ñ 0 and T Ñ 8. The former follows by looking at the definition of G T using the geometric side because by the prime geodesic theorem the number of prime closed geodesics of length less than x is Ope 2x {2xq (cf. Section 9); for the latter, it follows by looking at its definition using the spectral side, because the Weyl law implies that the number of spectral parameters less than t in absolute value is Opt 3 q (cf. Section 5), and p G is still a Gaussian. By the properties of the Mellin transform discussed in subsection 4.1, this implies that the numerator ş 8 0 G T T s dT T is an entire function of s. On the other hand, the integrand in the denominator T p GpT q is rapidly decaying for T Ñ 8, and by looking at the Taylor series at t " 0 one obtains an asymptotic expansion with only odd exponents. Hence the denominator admits an analytic continuation to the whole plane which is regular and non-zero at 0 because GpT qdT ą 0. The claim then follows by uniqueness of the analytic continuation.
The case of p 1 2a 1 r´a,as q˚k, with k ě 6 even is analogous. Assume for simplicity a " 1. First of all, the derivative of this function is regular enough for the odd trace formula to hold, see Section 3.5. Notice that in this case Looking at the spectral side, we see then that for T Ñ`8. Furthermore G T is still rapidly decaying for T Ñ 0, because G is compactly supported on the geometric side. The numerator is therefore a holomorphic function on the half-plane Repsq ă k´1. As in the case of Gaussians, T p GpT q has a Taylor expansion at 0 with only odd exponent terms, and The denominator is therefore a meromorphic function on Repsq ă k´1, regular and non-vanishing at zero, and we conclude as in the case of the Gaussian.
Let us unravel the statement of the above result in a way that is useful for concrete computations. If we choose a cutoff L ą 0, we have the formula (14) η sign " where at the numerator we evaluate the integral near zero using the geometric definition, and the integral near infinity using the spectral definition. Explicitly, for the eta invariant of the odd signature operator: and where we made a simple substitution in the integral.
Finally, the case of spinors follows in the same way by using the identity instead. The only additional observation is that the computation in (13) still holds even when the Dirac operator has kernel. In fact, we have GpT qT s dT as is the second row the parameters s n equal to zero do not contribute.

Effective local Weyl laws
In this section, we discuss upper bounds for the number of spectral parameters in a given interval for our operators of interest. These upper bounds are expressed purely in terms of injectivity radius and volume, and we refer to such bounds as local Weyl laws, see also [38].
To place this in context, recall the classical Weyl law: for every dimension n, there exists a constant C n such that for every Riemannian n-manifold pX, gq, the asymptotic (15) #teigenvalues λ of ∆ g with ? λ ď T u " C n¨v ol g pXq¨T n .
holds, where ∆ g is the Hodge Laplacian acting on functions. Analogous versions hold more generally for squares of Dirac type operators, such as the Hodge Laplacian pd`d˚q 2 acting on k-forms or the Dirac Laplacian D 2 B 0 , see [4]. From Equation (15), one expects the following local version of Weyl law to hold #teigenvalues λ of ∆ g with ? λ P rT, T`1su " n¨C n¨v ol g pXq¨T n´1 .
However, for our purposes it will be important not to just understand the asymptotic behavior of the number of the eigenvalues in a given interval, but also to provide effective upper bounds on it; in particular, our goal is to prove upper bounds of the form and similar bounds for the other operators we are interested in, namely the odd signature and Dirac operators. For our application, it is crucial to express the upper bound in (16) (and its analogues for all operators we study) uniformly in the parameter T, where the constants A and B are expressed explicitly in terms of the geometry of X. The leading constant A appearing in our upper bounds will generally be larger than the optimal one C 1 n¨v ol g pXq.
To prove these local Weyl laws, we will evaluate the trace formulas using even test functions of support so small that the sum over the closed geodesics vanishes. The estimates we will prove are effective in terms of volume and injectivity radius, but not of a nice form in terms of the input; for this reason, at the end of our computations we will specialize to the concrete case in which inj ą 0.15 and vol ă 6.5, e.g. the case of a manifold in the Hodgson-Weeks census. We will also assume throughout that b 1 " 0.
5.1. Preliminaries on test functions. Fix a value of R ą 0 (which will later be a given lower bound on the injectivity radius). Choose an even function ϕ with support in r´2, 2s and non-negative Fourier transform. Again, our convention is To simplify our discussion, we will make the following: Assumption. We will assume for simplicity throughout the section that both ϕ and p ϕ achieve their maximum at 0, and that the minimum m R of p ϕ in r´R, Rs is achieved at˘R.
This function satisfies the assumptions, and m R " 0.19643.
Remark 5.1. Of course, this implies that ϕ 0 satisfies the assumption also for R ă 0.15. To obtain a function that satisfies the assumption for large values of R, we can look at functions of the form ϕ 0 pCxq.

Local
Weyl law for coexact 1-forms. We begin with the case of coexact 1-forms, which is the simplest to analyze. For ν ě 0, denote by δ˚pνq the number of spectral parameters t j with |t j | P rν, ν`1s. We evaluate the trace formula in Theorem 3.2 for the test function ϕ R,ν pxq with R less than the injectivity radius of Y . As ϕ R,ν is supported in r´2R, 2Rs, and the injectivity radius is exactly half the length of the shortest geodesic of Y , the sum over the geodesics vanishes, and we have the identity volpY q 2π´ϕ By the identities of the previous subsection we therefore obtain volpY q πˆϕ p0q´1 R¨m R¨δ˚p νq using first that p ϕ R is non-negative, and then that for t P rν, ν`1s. For any choice of suitable test function ϕ, this provides the following upper bound on δ˚pνq in terms of volpY q and injectivity radius: here we used that p ϕ achieve its maximum at 0. Working out the computations using our test function ϕ 0 , we obtain the following. Remark 5.2. The bound obtained with this approach gets worse as R goes to zero; the same is true for the spectral density on functions and spinors. One way to obtain significantly better estimates in this case is to consider a Margulis number µ ą 0 for Y . Recall that for such a number, the set of points of Y with local injectivity radius ă µ{2 is the disjoint union of tubes around the finitely many closed geodesics with length ă µ. For example, in [33] it is shown µ " 0.1 is a Margulis number for all closed oriented hyperbolic three-manifolds. As the number of closed geodesics shorter than µ can be bounded above in terms of the volume [18], one obtains estimates for the spectral density by considering a test function supported in r´µ, µs. We will not pursue the exact output of this approach in the present work.

Local
Weyl law for eigenfunctions. The case of functions is more involved because of the possible appearance of small eigenvalues (i.e. the ones corresponding to imaginary parameters r j ). Set δpνq to be the number of parameters in rν, ν`1s, and let δ s be the number of small eigenvalues. We apply Theorem 3.1 choosing as before R to be less than the injectivity radius, again the sum over geodesics vanishes, and we get the identity ÿ z ϕ R,ν pr n q "´v olpY q 2π ϕ 2 R,ν p0q Here we recall that r n P R ě0 Y ir0, 1s corresponds to the eigenvalue λ n " 1`r 2 n . We start by bounding the number of small eigenvalues. For this purpose, we set ν " 0. For real t, we have which as a function of t P R is non-negative, even, convex, and has minimum at 0. Therefore we have´v To bound large eigenvalues, we look at z ϕ R,ν for ν ‰ 0. Unfortunately, this is not necessarily positive on the imaginary axis. On the other hand, we have for t P r0, 1s where the last inequality holds because pe 2Rt´e´2Rt q{t is increasing for t P r0, 1s. The trace formula then implies volpYq Concretely, using again the test function ϕ 0 , we obtain. so that volpY q πˆ1 4 ϕp0q´1 R 2 ϕ 2 p0q`ν 2 ϕp0q˙ě Specializing using the test function ϕ 0 , we obtain: Let Y be a hyperbolic rational homology sphere with volpY q ă 6.5 and injpY q ą 0.15. Then for any spin c structure, the inequality δ D pνq ď 18.7ν 2`2 561.3 holds.

Geometric bounds for the Frøyshov invariant
In this section we will use the local Weyl laws from the previous section to prove bounds on the eta invariants in terms of volume and injectivity radius. Combining this with Proposition 1.1, we will be able to prove Theorem 0.1. For simplicity of notation, assume that we have the inequalities where the specific values of the constants for a manifold with vol ă 6.5 and inj ą 0.15 were determined in the previous section.
6.1. Bounds on η sign . Recall from Subsection 4 (after setting L " 1): for an arbitrary admissible test function G, η sign " For our purposes, it is convenient to restrict our attention to Gpxq " p 1 2 1 r´1,1s q˚k for even k ě 6. This function has support in r´k, ks, and its Fourier transform is p Gptq " sinc k ptq where sincptq :" sinptq{t. We will denote c k :" Let us consider the numerator of the expression for η sign ; we will evaluate the first term using the geometric side of the trace formula and the second term using the spectral side. Starting with the second term, we have We split the last sum in two parts ř |tn|ď2`ř|tn|ą2 . In absolute value, the first part can be bounded above as For the second part, using sinc k pT q ď T´k, we have pApζpk´3q´1q`Bpζpk´1q´1qq and thereforěˇˇˇż For the geometric side, using a simple substitution, we have

Gp pγqq pγq
As pγq ě 2injpY q, we have by taking absolute valuešˇˇˇż To proceed, we notice that the right hand side is (up to a constant) the geometric side of the trace formula for functions, and we can provide bounds in terms of the spectral density of eigenfunctions. In particular, using Theorem 3.1, we have Gpr n q where we used that G 2 p0q is negative. We deal with imaginary and real parameters separately. For imaginary parameters, we have p Gpitq " sinc k pitq " sinh k ptq t k which is increasing in t and therefore we obtain the inequality ÿ imaginary rn p Gpr n q ď sinhp1q k¨δ s ď C¨sinhp1q k .
For the real parameters, using p G ď 1 and p G ď 1{x k respectively we have Hence, putting everything in §6.1 together, we havěˇˇˇż Putting everything together, we obtain the following.
Proposition 6.1. For every even k ě 6 the inequality olds, where we used the notation introduced above.
Choosing for example k " 8, and plugging in the constants we found in the previous section, we obtain the following effective estimate. Corollary 6.2. If Y is a hyperbolic rational homology sphere with volpY q ă 6.5 and injpY q ą 0.15, then |η sign pY q|ď 108267.
6.2. Bounds on η Dir . The discussion for the case of spinors is identical, with the final result obtained by substituting the constant D with E. This is because the quantity to take it is based on the identity Here we will again bound the integral ş 8 1 G T dT T using the bound for the spectral density δ D pνq, and the integral ş 1 0 G T dT T using the trace formula for functions. The final result is the following: Proposition 6.3. For every even k ě 6 the inequality olds for all spin c structures.
Setting again k " 8, we have the following effective estimate.

An explicit example: the Weeks manifold
In order to prove Theorem 0.1, we used test functions supported in r´2¨inj, 2¨injs, as the only geometric input we had about the length spectrum was the injectivity radius. On the other hand, in specific examples one can access very concrete information about the length spectrum, and therefore one can apply the trace formula to a much larger class of test functions. In turn, one can use this to provide explicit computation of Frøyshov invariants. In this section, we show how this approach can be implemented in a simple example of minimal hyperbolic L-space, the Weeks manifold W; a similar approach works also for other small volume hyperbolic minimal L-spaces in the Hodgson-Weeks census we discussed in [28].
Recall that H 1 pW, Zq " pZ{5Zq 2 . In our discussion, we will not discuss explicitly error bounds, to keep the section streamlined; we will deal with rigorous estimates of errors in our approximations when dealing with the Seifert-Weber dodecahedral space in the proof of Theorem 0.2. To this end, one can think of this section as both a warm-up exercise and a sanity check -the latter because the Frøyshov invariants of W can be computed directly with other purely topological methods. We have indeed the following. This is essentially showed in [36], using the fact that W is the branched double cover of the knot 9 49 [32]. The key point is that the 9 49 differs from an alternating knot only for an extra twist, and the authors showed that under favorable circumstances this allows to compute the Frøyshov invariants in terms of a Goeritz matrix for the knot (generalizing the method of Ozsváth and Szabó [41], which applies to alternating knots).
Going back to our spectral approach, the signature eta invariant was computed in [16] to be η sign " 0.040028711 . . . 3 We compute the eta invariant for the Dirac operators using our explicit expression from Section 4. In order to do so, we first compute the spin c length spectra up to cutoff R " 6.5 using the algorithm described in Section 2 (applied to data obtained from SnapPy). We then take the approach from [28] and use the even trace formula to obtain information about the spectrum. More specifically, using ideas of Booker and Strombergsson, for each spin c structure s we determine an explicit function with the property that if˘s are eigenvalues of the Dirac operator whose multiplicities add to m, then J s p|s|q ě m. The pictures in the range r0, 6s can be found in Figure 1; Class 1 is the class of the spin structure, and the remaining 24 spin c structures can be grouped in four groups (each consisting of six elements) with identical picture.
Remark 7.1. In fact, this is a consequence of the action of the isometry group D 12 on the set of spin c structures [32]. We will exploit symmetry under the isometry group in the more complicated case of SW in Section 10.
We compute an approximation to η Dir via the formula (14) taking G to be a Gaussian, see Lemma 4.1. In this process, we only consider the sum over geodesics with length ď 6.5, and only sum over eigenvalues where a precise guess can be made. More specifically: ‚ In the Classes 1, 4, 5, we consider the contribution of the smallest eigenvalue as suggested by the picture (and consider the rest of the spectrum as an error term). Let us point out that can in principle use the method of [29] to actually prove that an eigenvalue with the right multiplicity in the tiny window suggested by the picture; here we also use the fact that in the spin case eigenvalues always have even multiplicity, because the corresponding Dirac operator in quaternionic linear. The sign of the eigenvalue can also be determined by using the odd trace formula in Theorem 3.3. ‚ In Classes 2 and 3, where a precise value for the first eigenvalue is not available, we simply consider the whole spectrum to be an error term, and take a more dilated Gaussian (which has a narrower Fourier transform) to make such error smaller. This In order to make this approximate computation into an actual proof of Proposition 7.1, we need to provide estimates on the terms we did not take into account in the sum. Let us discuss the various steps: (1) We know that the Frøyshov invariants are rational numbers, and in fact information about their fractional part can be obtained by purely topological means. For example, in the case of Weeks, one can show a priori that all Frøyshov invariants have the form p2n`1q{10. Given this, one only needs to prove that the errors are less than 1 20 in order to conclude. In fact, by taking into account additional topological information coming from the linking form, one only needs to prove much weaker estimates for the error; this will be crucial in our proof of Theorem 0.2.
(2) To bound the error resulting from truncating the sum over geodesics, we need to have a good understanding of the geodesics with length in a certain interval; in a specific example, this can be done effectively using the trace formula for functions (in the same spirit as the Prime geodesic theorem with errors [10]). The key point is then to get reasonable bounds on the spectral density (sharper than those in Section 5). While we will not pursue the details of this approach here for the Weeks manifold, we will do so in subsequent sections for the more challenging case of the Seifert-Weber manifold in order to prove Theorem 0.2. In particular, each of the next three sections will address one of the three aspects discussed above. Notice that in the case of p2q and p3q, the error estimate can be improved by computing a larger portion of the (spin c ) length spectrum. for the unique spin structure on the Weeks manifold.

The linking form of the Seifert-Weber dodecahedral space
In this section we compute the linking form of the Seifert-Weber dodecahedral space SW. This will be the key (and only) topological input for our computation of its Frøyshov invariants, and will give us concrete information on their fractional parts. Recall that for oriented rational homology three spheres, the linking form is the bilinear form Q : H 1 pY, ZqˆH 1 pY, Zq Ñ Q{Z defined geometrically as follows: given elements x, y, choose n for which ny " 0 and pick a chain T with BT " ny. Then Qpx, yq is the intersection number of x and T , divided by n. More abstractly, the corresponding map where the first isomorphism is Poincaré duality and the second is the Bockstein homomorphism for the short exact sequence of coefficients In particular, Q # is an isomorphism and Q is non-degenerate. . Remark 8.1. In fact, one can show that (up to scalar) every non-degenerate form has that shape with respect to some basis. But our point is that we also can identify the basis a, b, c geometrically in this specific case. Our computation also suggests that the linking form of a hyperbolic three-manifold is computable by taking a Dirichlet domain with face pairing maps as input, so that the approach to the computation of the Frøyshov invariants of SW we will discuss in the upcoming sections works more generally.
Proof. Recall that SW is obtained from a dodecahedron by identifying opposite faces by 3{5 of a full counterclockwise rotation. The identification is described explicitly in [17], see Figure  2. We have that: ‚ the 20 vertices are identified to a single point; ‚ the 30 edges are identified in 6 groups of 5 elements, and we denote the corresponding generators of homology a, b, c, d, e, f . ‚ the 12 faces are identified in pairs as follows: 1 Ø 12, 2 Ø 9, 3 Ø 10, 4 Ø 11, 5 Ø 7, 6 Ø 8.
The six pairs of faces give us relations between the generators of the first homology. We will denote the corresponding chain by F i , where i is the smallest of the two indices, and orient it with the opposite orientation as the one inherited as a subset of the plane. We get: Simple elementary row operations reduce this system to d " a`2b`3c e " 3a`2b`c f " 3a`4b`3c and (17) 5a " 5b " 5c " 0 We therefore see that a, b, c generate the first homology H 1 pSWq -pZ{5q 3 . Furthermore, we can interpret the equations (17) as the geometric identities 5a " BpF 1´F2`F3`F5´F6 q 5b " BpF 1´F2´F3`F4`F6 q 5c " BpF 1`F2´F3´F4`F5 q.
To compute the linking form it is convenient to introduce a second basis of the homology: let A, B, C be the generators corresponding to the oriented segments connecting the centers of the faces 12 to 1, 9 to 2 and 10 to 3 respectively. Using Figure 2, it is easy to compute that at the level of homology For our choice of orientation, we have that A has intersection`1 with F 1 , and 0 with the other pairs; the analogous statement holds for B and F 2 and C and F 3 respectively. Using the geometric descriptions of chains bounding for 5a, 5b and 5c above, we therefore get the following values for the linking numbers between elements in A, B, C and a, b, c.
Finally, a simple change of basis to express everything in terms of the basis a, b, c concludes the proof.
A useful observation for what follows is that Q SW px, xq for x ‰ 0 attains possible values as in the following table.
Q SW px, xq number of x ‰ 0 that attain the value 0 24 1/5 30 2/5 20 3/5 20 4/5 30 Another useful observation can be made by looking at the isometry group of SW. Recall that the latter is isomorphic to the symmetric group S 5 , and acts faithfully on the first homology group [31]. In fact, from the description in [31] we see that the natural map IsompSWq ãÑ OpQ SW q has image contained in SOpQ SW q. The latter has 120 elements, and we get therefore an isomorphism IsompSWq -SOpQ SW q. We have therefore the following. Proof. This follows by the Witt extension theorem [45, Theorem 1.5.3]: because Q SW is non degenerate, given two elements x, y ‰ 0 satisfying Qpxq " Qpyq, there is A P SOpQ SW q for which Ax " y.

Effective estimates for closed geodesics on the Seifert-Weber dodecahedral space
For practical purposes, the most effective way to evaluate the eta invariants using an expression such as (14) is to use a suitably dilated Gaussian (as demonstrated by the precise computations for the Weeks manifold in Section 7). The main complication is that the function is not compactly supported, and therefore we need to provide effective bounds on the tail sums ÿ pγqěR pγ 0 q¨2 sinpholpγqq for the signature case and ÿ pγqěR pγ 0 q¨2 sinpholpγqq¨cospϕγq in the Dirac case respectively. Here L is the parameter at which we split the integral defining the eta function, and R is the cutoff for geodesics (later on we will take them to be 1.7 and 7.5 respectively). For these purposes, we need a good understanding of the behavior of the lengths of geodesics, and more specifically of the quantity ÿ pγqPrT´1{2,T`1{2s pγ 0 q. naturally appears when studying the asymptotic number of prime geodesics, and in particular it plays a key role in the proof of the prime geodesic theorem #tprime geodesics γ with pγq ď T u " e 2T {2T, see for example [14]. In fact, under the heuristic correspondence between closed geodesics on a hyperbolic manifold and prime numbers (under which e pγq is the analogue of p), the quantity (18) corresponds to the Chebyshev function ψpxq " ÿ p k ďx logppq.
It is well known [1, Chapter 4] that the prime number theorem #tprime numbers p ď xu " x{ logpxq is equivalent to the asymptotic ψpxq " x.
Precise asymptotics for (18) can be obtained by evaluating the trace formula for functions in Theorem 3.1 for a smoothed version of the function coshptq1 r´T,T s (see also [10] for the case of hyperbolic surfaces). In our case, in order to obtain reasonable effective constants, we will use instead combinations of Gaussian functions. Furthermore, while the general results of Subsection 5.3 apply to our case, we can obtain considerably sharper bounds for the spectrum using as input our knowledge of the length spectrum of SW up to cutoff R " 8 (rather than just the injectivity radius), as in [29]. Furthermore, we have computed the spin c length spectrum of SW for all spin c structures up to cutoff R " 7.5 using the method described in Section 2.
9.1. The spectral gap for the Laplacian on functions. The first step is to understand the spectral gap for the Laplacian on functions -as we saw in Subsection 5.3, small eigenvalues are the main cause of large error estimates. In our case, we have.
Proposition 9.1. The Laplacian on functions for SW has no small eigenvalues, and the smallest parameter satisfies r 1 ą 2.8, corresponding to λ 1 ą 8.84.
Remark 9.1. The absence of small eigenvalues in the statement of Proposition 9.1 was to be expected, as it is a consequence of the generalized Ramanujan conjecture for GL 2 . See for example [6].
The proof of this proposition is based on the Booker-Strombergsson method applied to the trace formula for functions (which is indeed, in the case of surfaces, the original setup of their approach [7]). The main difference is that we now get two pictures, one for the imaginary parameters ( Figure 3) and one for the real ones (Figure 4) 4 . In both cases, to exclude the parameter t, we minimized the quantity Hpr n q over the same space of functions used in [28], subject to the constraint p Hptq " 1, using the trace formula in Theorem 3.1. Here the cutoff is R " 8, as in [29]. Notice that r 0 " i is always a spectral parameter (corresponding to λ 0 " 0), and is not included in the sum (19). 9.2. Spectral density on functions. We now use the spectral gap and our explicit knowledge of the length spectrum to provide refined bounds on the spectral density. These two extra ingredients will allow us to greatly improve the estimates from Section 5. Let us apply the trace formula to which is the same kind of function we used in Section 5. In this case, we havé We will use the fact that that sinpxq x˙6 ě 0.777 for x P r´0.5, 0.5s. To prove upper bounds, we will look again at the trace formula. In our setup, we can compute explicitly an upper bound for the sum over geodesicšˇˇˇˇÿ because H 0 is supported in r´6, 6s and we know the length spectrum up to cutoff R " 8. Furthermore, recalling that |sinpi`νq| 2 " cos 2 pνq sinh 2 p1q`sin 2 pνq cosh 2 p1q " sinh 2 p1q`sin 2 pνq, we have that the contribution of the zero eigenvalue is bounded above by We are interested in upper bounds for the number of spectral parameters in rν´1{2, ν`1{2s for T ě 3. The quantity (20) is bounded above by 0.0055 for ν ě 3. Putting everything together, and recalling that SW has volume about 11.19, we get the following refined local Weyl law. Here c ą 0 is a parameter to be determined later. Let us discuss its value at the various terms in the trace formula, starting from the spectral side. The contribution of the zero eigenvalue (corresponding to r 0 " i) is 2 c 2π c e 1{2c coshpT q.
Using Proposition 9.1, the contribution of the real parameters is bounded above (independenty of T ) by The identity contribution is (this is negative provided T 2 ą 1{c). All these terms are explicitly computable, and we will use this information to provide upper bounds on ÿ lpγqPrT´1{2,T`1{2s pγ 0 q.
as follows. Notice that |1´e C pγq ||1´e´C pγq | ď pe `1qp1`e´ q " 2 coshp q`2 and F c,T p q ě e´c {8 for P rT´1{2, T`1{2s We therefore have pγqPrT´1{2,T`1{2s pγ 0 q. and we can use the explicit upper bound for the first expression, obtained by combining the trace formula with the estimates (21) and (22) , to get an upper bound for ř pγqPrT´1{2,T`1{2s pγ 0 q, depending on a parameter c. We see empirically that we obtain the best estimate for c " 5, and we have the following: where ApT q is an explicit expression in T readily obtained from the discussion above.
We do not write explicitly the explicit expression of ApT q as it is quite long and not particularly illuminating, but we remark that the leading term is of the form C¨e 2T . 9.4. The geometric error for η. Finally, we will conclude by bounding the error in the evaluation of the eta invariant using the standard Gaussian Gpxq " e´x 2 {2 in the formula (14), and evaluating the geometric side up to length R " 7.
We can break the sum in two parts, one taking into account pγq P r7.5, 8s and one for pγq ě 8. The first part can be computed explicitly because we know the (standard) length spectrum of SW up to cutoff R " 8. For the second part, noticing that |1´e C pγq ||1´e´C pγq | ě pe ´1qp1´e´ q " 2sinhp q´2, pγqPrn,n`1s where A is the quantity from Proposition 9.3. Putting the pieces together, we obtain an estimate on the total error for the truncated sum depending on the choice of the parameter L. It is clear that the larger the value of L is, the better our estimate for the error in the truncated sum will be; on the other hand, we will see in the next section that larger values of L lead to significantly worse errors coming from the spectral side of the formula for the eta invariant in (14). We empirically found that the value L " 1.7 provides a very good bound for the sum of these two errors in our situation. In this case, for the geometric side we have the following.
Proposition 9.4. When evaluating the η invariant for either the signature or the Dirac operator using a standard Gaussian Gpxq " e´x 2 {2 , splitting point L " 1.7 and length cutoff R " 7.5, the error coming from truncating the sum on the geometric side is bounded above by 0.0376. 5

The Frøyshov invariants of the Seifert-Weber dodecahedral space
In this section, we finally prove Theorem 0.2. Our proof is based on the fact that the Seifert-Weber is a minimal hyperbolic L-space [29]. More generally, the approach of this final section can in principle be adapted to any minimal hyperbolic L-space, provided the linking form and a good portion of the length spectrum have been computed. Our result will be obtained by combining the Chern-Simons computations obtained via SnapPy, together with computations involving the trace formula. In principle, this result can be obtained directly via Snap [16]; unfortunately, we were not able to run the software on our laptops. Also, the proof provides a good example of our technique, in a simpler setup than the Dirac case (where our understanding of the spectrum is less precise).
Proof. Snappy computes the Chern-Simons invariant of SW to be cs "´0.033333 . . .

The relation
3η sign " 2cs`τ mod 2Z holds [3], where τ , which is the number of 2-primary summands in H 1 pSW, Zq, is zero in our case. Thus (23) η sign "´0.022222¨¨¨`2 3 n for some n P Z. Therefore, approximating the eta invariant to within error less than 1 3 pins down its value to high accuracy. Approximating η sign well is straightforward because we have a good understanding of the small coexact 1-form eigenvalues on SW (see Figure 5, which we used in [29] to show λ1 ą 2): ‚ The smallest eigenspace corresponds to (24) a λ1 P r1.42787720680237, 1.43033743858337s and has multiplicity exactly 6 (see [29]). Furthermore, using the odd trace formula one readily shows that this eigenvalue is positive. ‚ The next spectral parameter is larger than 2.
We compute an approximation of the eta invariant using Gpxq " e´x 2 {2 and L " 1.7 in (14), where ‚ we truncate the geometric sum at R " 7.5; ‚ in the spectral sum we approximate the first eigenvalue (with multiplicity 6) with the midpoint Ć a λ1 of the interval in (24), and consider the remaining part of the spectral sum as an error term. As Gptqdt " π, we have the approximation which is very close to the expression (23) for n " 2. All we need to do is provide an error bound on our computation. The error arising from truncating the geometric sum at R " 7.5 is bounded above by 0.0376 by Proposition 9.4, while the error arising from approximating the value of the first eigenvalue is bounded above by 6 π¨ż To estimate the error arising from truncating the spectral sum at the first eigenvalue, we can refine the estimates from Section 5 (which only involved volume and injectivity radius) because we have a direct knowledge of the length spectrum. In particular, we can use the test function cospT xq¨ˆ1 2δ 1 r´δ,δs˙˚6 where δ " 8{6 and compute explicitly the value of the sum over geodesics (as we know it up to cutoff R " 8), and use it to provide upper bounds on the number of eigenvalues in specific intervals. In particular we get: ‚ there are at most 22 spectral parameters in r2, 3s, and each contributes at most ‚ there are at most 21 spectral parameters in r3, 4s, and each contributes at most 1 π ş 8
‚ it is clear that the contribution of larger eigenvalues is negligible, as their number grows quadratically but the contribution decays superexponentially.
Taking all this into account, we obtain an error of at most 0.054, and the result follows. Figure 5. The function J SW ptq for coexact 1-forms, computed with cutoff R " 8, cf. [29].
Remark 10.1. As the numerical result suggests, η sign is a rational number. This follows because the invariant trace field Qp a´1´2 ? 5q of SW is CM-embedded, see [39]. In fact, one can show that η sign " 59{45 using the multiplicativity of the Chern-Simons invariants under covers, and the fact that SW admits a 60-fold cover with an orientation reversing isometry (for which η sign is therefore 0). 10.2. The spin structure. We now focus on the computation of hpSW, s 0 q where s 0 is the unique spin structure on SW. This is the simplest spin c structure to handle becausé 2hpSW, s 0 q " η sign {4`η Dir " n{4 for some n P Z. To see this, recall the classical theorem that every spin three-manifold bounds a compact spin four-manifold pX, s X q; in fact, we can choose X to be simply connected [21, Chapter VII] (this will be useful later). Then, looking at the expression of the absolute grading (3) modulo integers, we have that as grpX˚, s X , rbsq is an integer (it is the expected dimension of a moduli space) and c 1 ps X q is a torsion class, and the claim follows. Therefore, we only need to approximate η Dir with a good error. The spectral picture can be found in Figure 6. Here, we computed the spin length spectrum up to cutoff R " 7.5 using the algorithm described in Section 2.
For a spin structure, the Dirac operator is quaternionic and therefore all eigenspaces have even multiplicity. We can therefore conclude that |s 1 | ě 1.45. We compute an approximation to the eta invariant using Gpxq " e´x 2 {2 and L " 1.7 by truncating the geometric sum at R " 7.5 and considering the whole spectral side as an error term. We obtain 1.7q 2 " 0.641083369621 . . . and therefore the approximate valué 2hpSW, s 0 q « 0.968861147398778 This shows hpSW, s 0 q "´1{2 provided we can bound the error term for η Dir effectively. Again, the error from truncating the sum on the geometric side is bounded above by 0.0376 by Proposition 9.4. For the spectral side, we bound the number of spectral parameters as in the proof of Proposition 10.1 using the even spinor trace formula for the function (25) cospT xq¨ˆ1 2δ 1 r´δ,δs˙˚6 where δ " 7.5{6 (whose geometric side we can compute as it is supported in r´7.5, 7.5s). We get that there are at most 8 spectral parameters in r1.45, 2s, and each contributes at most 1 π ż 8
Furthermore, there are at most 12 parameters in r2, 3s and 31 parameters in r3, 4s; larger parameters are again negligible. Putting everything together, we see that the error is bounded above by 0.1564. Therefore, we havé 2hpSW, s 0 q P r0.812, 1.126s and the result in Theorem 0.2 follows because 1 is the only number of the form n{4 in that interval.
10.3. Conclusion of the proof. We now complete the proof of Theorem 0.2. The arguments in this section will be quite ad hoc because the spin c structures have a smaller spectral gap than the spin case, and in particular the error bounds we will be able to provide will not be as good as in the spin case. Of course, one could in principle obtain better bounds by computing larger portions of the length spectrum, but this is a quite challenging task from the computational viewpoint (compare with the discussion in [28]).
A corollary of the computation of the previous section is the simply connected spin manifold pX, s X q bounding SW we fixed satisfies 1 4 p´2χpXq´3σpXq´2q P Z As H 3 pX, BX; Zq " H 1 pX, Zq " 0, the restriction map H 2 pX, Zq Ñ H 2 pBX, Zq is surjective; therefore every spin c structure s " s 0`x on SW extends to a spin c structurẽ s " s X`x on X. As c 1 psq equals 2x up to torsion elements, we have As H 1 pX, Zq " 0, we havex 2 " lkpx, xq mod Z hence we conclude, 2hpSW, sq " lkpx, xq mod Z.
By the computations of Proposition 8 we know exactly the number of spin c which are not spin and for which the self-linking number equals a given value in`1 5 Z˘{Z. Also, we know from Corollary 8.2 that the isometry group acts transitively on the set of non-spin spin c structures with fixed self-linking number, so we expect a priori to only have 5 possible spectral pictures for the remaining spin c structures. This is indeed the case, see Figure 7.
Comparing the number of spin c structures with a given linking form, we conclude the following.
Lemma 10.2. The following statements hold: (1) for the spin c structures in Class 2 and 3,´2h has fractional part 1{5 and´1{5, or vice versa; (2) for the spin c structures in Class 4,´2h has fractional part 0;  As in the case of the spin structure, we compute an approximation to the Dirac eta invariant using Gpxq " e´x 2 {2 and L " 1.7 by truncating the geometric sum at R " 7.5 and considering the whole spectral side as an error term. The results for the approximated value of´2hpY, sq " η sign {4`η Dir are as in the following We then again need to bound errors, as in the case of the spin structure. The error from truncating the sum on the geometric side is bounded above by 0.0376, see Proposition 9.4. For the spectral side, using the even spinor trace formula for the function (25) we can provide decent bounds on spectral parameters on any given interval. Using this, we can conclude; each of cases requires a slightly different argument, and we treat them separately.
Proof of Theorem 0.2 for Classes 2, 3 and 4. We begin with Class 4, which is the most involved because it has the smallest spectral gap. Using our refined estimates on the number of spectral parameters as in the previous sections using the test functions (25), we see that there are at most 1 eigenvalue in r0.545, 1.05s, 2 in r1.05, 1.39s, 4 in r1.39, 1.85s, 7 in r1.85, 2.2s, 15 in r2.2, 3s, 30 in r3, 4s, for a total error of 0.6279. We therefore concludé 2h P r0.017, 1.274s so that is´2h " 1 in this case, as it is an integer by Lemma 10.2.
For Classes 2 and 3 the gap is wider, so it is easier to get good bounds. For Class 2 we can obtain for example´2 h P r´0.197, 0.784s. Because the fractional part is either´1{5 or 1{5, we obtain that´2h " 1{5. This in turn implies that for Class 3,´2h has fractional part´1{5; as we obtain with the same approach 2h P r´0.614, 0.347s, we conclude that´2h "´1{5 in this case.
The case of classes 5 and 6, which have fractional part 2{5 and´2{5 or vice versa, is more delicate because for Class 5 we can only prove a very small spectral gap. We first obtain more information on the fractional parts as follows. Proof. The output of our computations described in Section 2 provides the description of a spin c structure as a homomorphism where the natural basis is different from the one used in Section 8. It is easy to check that the linking form in this new basis is given by Indeed, this is the unique symmetric matrix that takes the correct values for the spin c structures of class 2, 3 and 4. The lemma is then proved by plugging in the explicit values for classes 5 and 6.
Proof of Theorem 0.2 for Class 6. For Class 6, we compute using again the test functions (25) that there are at most 4 parameters in r1, 1.5s, 5 in r1.5, 2s, 16 in r2, 3s and 30 in r3, 4s. We therefore have a total error of at most 0.459, and thereforé 2h P r0.083, 1.002s hence by the previous Lemma we have´2h " 3{5.
Finally, to deal with the case of Class 5, we need some additional information about the small eigenvalues. and has multiplicity 1. Furthermore, |s 2 | ą 1.55.
Proof. Using the same method as above involving the test functions (25) we conclude that there are at most two eigenvalues in this interval.
To prove that there is exactly one, we apply the trace formula to the Gaussian e´p x{1.7q 2 {2 by truncating the sum over geodesics at R " 7.5. The computations of the error bound involves the exact same quantities as in the proof of Proposition 9.4, and we obtain the following estimate for the sum over the spectrum: ÿ sn e´p 1.7¨snq 2 {2 P r0.9678, 1.0789s. Now, for t P r0.0408361, 0.4077692s we have e´p 1.7¨tq 2 {2 P r0.786417, 0.9976s and the spectral picture shows that eigenvalues which are not in this interval are ě 1.55. The same trace formula argument used seversal times above to prove upper bounds on spectral density shows that there are at most 6 parameters in r1.55, 2s, 16 in r2, 3s, 30 in r3, 4s, from which we see that the contribution of the eigenvalues ě 1.55 has to be at most 0.2432. Using the fact that there are at most two eigenvalues in the interval, we conclude by direct inspection that there has to be exactly one.
Proof of Theorem 0.2 for Class 5. For our purposes the main complication arises from the fact that it is hard to determine the sign of the small eigenvalue using the trace formula. This is because in order to study the sign we need to use odd test functions, and these will be small near zero. We will therefore study the two possibilities separately. We approximate the contribution of the small eigenvalue with the average 0.7157945 of 1 π ?
which introduces an error of at most 0.229992. The total error is computed to be then at most 0.3289 (luckily, |s 2 | is large). If the small eigenvalue s 1 were negative we would havé 2h P r´0.588, 0.069s which does not contain any number with fractional part 2{5. Therefore s 1 ą 0, and we havé 2h P r0.8432, 1.5011s from which we conclude´2h " 7{5.
Appendix A. Generalities on the group G and the trace formula All the trace formulas we will need in the present paper are obtained by specializing the general trace formula for a cocompact torsion-free lattice r Γ in the Lie group G " tg P GL 2 pCq : | detpgq| " 1u.
The strategy of the proof follows very closely that of [28,Appendix B], where we studied the case of PGL 2 pCq: (1) express the very general trace formula in terms of irreducible representations and geometric data; (2) understand the representation theoretic incarnation of the differential operators under consideration; (3) choose suitable test functions to isolate the relevant representations. In [28, Appendix B], we tackled the first step for PGL 2 pCq: we derived a general trace formula for cocompact lattices in the group PGL 2 pCq (which was called G in the reference). So as not to repeat ourselves, we simply state the trace formula for G and highlight the differences between it and the very similar trace formula from [28].
Remark A.1. As it will be clear from the discussion, this approach involving G is only strictly necessary when dealing with spinors; the trace formulas for functions and forms, both even and odd, can be derived from the general trace formula for PGL 2 pCq.
A.1. Notation for subgroups of G. We will work with the maximal torus T Ă G of diagonal matrices; we will use the parametrizations (27) T " "ˆe u{2 e iα 0 0 e´u {2 e iβ˙: u P R, α, β P R{2πZ * and (28) The center Z of G is the diagonal copy of U 1 . We will equip T with the Haar measure (29) dt " du dα 2π dβ 2π .
The Weyl group W has two elements, and is generated by pu, α, βq Þ Ñ p´u, β, αq.
Remark A.2. Using this notation, the complex length of the image in PGL 2 pCq of an element of T is u`i2θ.
We will denote by B the subgroup of upper triangular matrices; the modular function is then Finally, K " U 2 is the maximal compact subgroup; we can identify G{K " PGL 2 pCq{P U 2 " H 3 using the upper half-space model H 3 " CˆR ą0 . The tangent space at p0, 1q is then identified with p 0 " isup2q. We will also denote p " p 0 b C for its complexification.
A.2. Unitary representations of G. Every irreducible unitary representation of G is isomorphic to one of the following: (a) the one dimensional representation with character det k for some k P Z.
Again, there are some coincidences among these representations.
Proof. The proof is very similar to the case of PGL 2 pCq covered in [28,Appendix B]. For every smooth compactly supported test function f on G, tracepπ s 1 ,s 2 ,n 1 ,n 2 q " x Sf pχ´1 s 1 ,s 2 ,n 1 ,n 2 q, where we view Sf as a W -invariant function on the diagonal torus T. By the main theorem of Bouaziz [9], the Satake transform f Þ Ñ Sf maps onto the W -invariant compactly supported smooth function on T. Therefore, two representations π s 1 ,s 2 ,n 1 ,n 2 and π t 1 ,t 2 ,m 1 ,m 2 are isomorphic iff p F pχ s 1 ,s 2 ,n 1 ,n 2 q " p F pχ t 1 ,t 2 ,m 1 ,m 2 q for all smooth, compactly supported functions F on T which is W -invariant, i.e.
A.3. The trace formula for G. We have the following general result.
holds. Here the second derivatives in v and θ are taken with respect to the parametrization (28) of T , and |W | " 2.
This is in fact a generalization of the trace formula from [28], which can be recovered by using test functions on the diagonal torus T invariant by the center Z of G. The proof of this result is essentially identical to the one of the formula in [28], so we will focus on pointing out similarities and differences between Proposition A.2 and the trace formula from our first paper.
‚ The "identity contribution" to the trace formula has a contribution for each element of r Γ X Z. The group Z is compact so this sum is finite. In our case of interest, r Γ is the lift of a lattice Γ Ă PGL 2 pCq, and r Γ X Z " t1u. ‚ Recall that the function F pu, α, βq is W -invariant iff F pu, α, βq " F p´u, β, αq.
In particular, we will use F pu, α, βq " K˘pα, βq¨H˘puq, where K˘are symmetric and antisymmetric respectively and H˘are even and odd respectively, define W -invariant functions on T. ‚ The universal constant´1 8π can be determined in several ways; for example, by specializing this formula in the case of the even trace formula for coexact 1-forms and comparing it to the one from [28].
representation theory of PGL 2 ; in particular, we identified coexact 1-eigenforms in terms of isotypic vectors in the irreducible representations, and the corresponding eigenvalue using the Casimir eigenvalue. We will do the same here, where we consider the maximal compact subgroup K " U 2 . In particular we show the following two results. 6 Proposition B.1. Eigenforms of˚d correspond to p-isotypic vectors where p " p 0 b C. An irreducible unitary representation of G contains a p-isotypic vector if and only if it is isomorphic to π s 1 ,s 2 ,1,´1 with s 1 , s 2 purely imaginary. Furthermore, each copy of π s 1 ,s 2 ,1,´1 Ă L 2 p r ΓzGq contains exactly a one dimensional eigenspace of˚d, spanned by an eigenform with eigenvalue ips 1´s2 q{2.
Recall that a p-isotypic vector in a G-representation π is a non-zero element of Hom K pp, πq. Notice that as the Hodge Laplacian on coexact 1-forms is the square of˚d, this result implies (after changing the parametrization) those in [28]; in that case we computed the corresponding eigenvalue via Kuga's lemma. Notice that while this result can be proved directly using the group PGL 2 , the following one relies essentially on the larger group G.
Proposition B.2. Eigenspinors correspond to S _ -isotypic vectors where S _ is the dual of the standard representation of K " U 2 on C 2 . An irreducible unitary representation of G contains an S _ -isotypic vector if and only if it is isomorphic to π s 1 ,s 2 ,´1,0 with s 1 , s 2 purely imaginary. Furthermore, each copy of π s 1 ,s 2 ,´1,0 Ă L 2 p r ΓzGq contains exactly a one dimensional eigenspace of the Dirac operator, spanned by an eigenspinor with eigenvalue ips 1´s2 q{2. B.1.1. The invariant connection on G Ñ G{K. We follow the notation of Subsection A.1. Note that G π Ý Ñ G{K (with K acting on the right by multiplication) is a principal K-bundle. The assignment H g :" pL g q˚p 0 Ă T g pGq defines a subbundle H Ă T G. Observe that (1) The projection H g Ñ T g pGq πÝ Ñ T gK pG{Kq is an isomorphism. 6 Recall that our convention is that orms a positively oriented basis of p0 " isup2q.
(2) The subbundle H is right K-invariant: pR k q˚H g " pR k q˚pL g q˚p 0 " pL g q˚pR k q˚p 0 because left and right multiplications commute " rpL g q˚pL k q˚s " pL k q´1 pR k q˚‰ p 0 " pL g L k q˚pconjugation by k´1q˚p 0 " pL gk q˚Adpk´1qp 0 " pL gk q˚p 0 because p 0 is invariant under AdpKq (1) and (2) imply that H defines a connection on G Ñ G{K. Item (3) implies that H is left G-invariant.
The covariant derivative ∇ V on F V associated to the connection H Ă T G is given by where r X denotes the horizontal lift of X to T G via the invariant connection H from §B.1.1. Via the trivialization p 0ˆG Ñ H pX, gq Þ Ñ pL g q˚X from §B.1.1, we may regard ∇ V s P Γ`F V b Ω 1 pG{Kq˘, which is the bundle associated to the representation V b p _ 0 » Hompp 0 , V q of K, as the function ∇ V s : G Ñ Hompp 0 , V q (30) g Þ Ñ pX Þ Ñ dsppL g q˚Xq " Xps˝L g qq .
Let r denote the representation ote that ∇ V s, as defined by formula (30), satisfies the K-equivariance propertỳ B.1.3. Dual picture for sections of F V and the covariant derivative ∇ V . For our purposes it will be convenient to rephrase everything in terms of a dual picture, which is more manifestly representation theoretic. There is a canonical isomorphism The inverse mapping pΦ V q´1 is most easily expressed using a basis tv i u for V and its corresponding dual basis tv _ i u for V _ : The map Φ :" Φ V satisfies the following equivariance properties: (1) ΦpL g sq " L g Φpsq and ΦpR g sq " R g Φpsq. In C 8 pG, V q, L g spxq " spgxq, R g spxq " spxgq and on HompV _ , C 8 pGqq, the corresponding actions are (2) Φ carries the subspace ΓpF V q " ts : G Ñ S : spgkq " ρpkq´1spgqu to the subspace Hom K pV _ , C 8 pGqq, where K acts by right translation on C 8 pGq and by ρ _ on V _ .
(3) Upon combining (30) with the explicit formula for Φ´1 from (32), we see that Φ intertwines the covariant derivative ∇ V with operator there is a commutative diagram The above restricts to a commutative diagram (4) K-equivariant maps α : V Ñ W induce maps ΓpF V q Ñ ΓpF W q (by post-composition with α) and on Hom K pV _ , C 8 pGqq Ñ Hom K pW _ , C 8 pGqq (by pre-composition with α _ ). These induced maps commute with the isomorphism Φ. ΓpF V q " ts : r ΓzG Ñ S : sp r Γgkq " ρpkq´1sp r Γgqu which is identified with, via the isomorphism Φ V , with Hom K pV _ , C 8 p r ΓzGqq.
The descended covariant derivative ∇ V and its incanation Θ V in the dual picture admit the "same formulas mod r Γ". In particular, and there is a commutative diagram Given a K-equivariant map α : V Ñ W, we denote the associated bundle map (obtained by post-composition by α on ΓpF V q or by precomposition by α _ in the dual picture) by α. We also have commutative diagrams Because ∇ V is built from the derivative of the right translation action of G on C 8 p r ΓzGq, the operators Θ V are amenable to representation-by-representation analysis. Likewise, the bundle maps α are amenable to representation-by-representation analysis. For irreducible (unitary) representations π of G, denote by π 8 Ă π the dense subspace of smooth elements. We define the following π-isotypic versions of the operators Θ V and α: and α π : Hom K pV _ , π 8 q Ñ Hom k pW _ , π 8 q T Þ Ñ T˝α _ .
Given a subrepresentation π Ă L 2 p r ΓzGq, the above π-isotypic operators fit into commutative diagrams (34) where the vertical maps ι in both diagrams are induced by the inclusion π Ă L 2 p r ΓzGq.
B.2. Automorphic incarnation of the spinor bundle and Dirac operator. Denote as before by S be the standard representation of U 2 on C 2 .
Definition B.1. Let r Γ Ă G be a lattice lifting the closed hyperbolic 3-manifold lattice Γ Ă PGL 2 pCq. The spinor bundle over ΓzH 3 " r ΓzH 3 " r ΓzG{K associated to the lift r Γ Ă G of Γ is the vector bundle F S . It comes equipped with the covariant derivative ∇ S .
B.2.1. Clifford multiplication on F S . The linear map C : S bp 0 Ñ S given by 2i times matrix multiplication is U 2 -equivariant: In what follows, we will make use of the (physicists') Pauli matrices Remark B.1. In the hyperbolic metric, the Pauli matrices σ 1 , σ 2 , σ 3 are orthogonal to each other, but have norm 2. To see why the latter holds, notice that e tσ 3 "ˆe t 0 0 e´t˙.
In the upper half-space model, this maps p0, 0, 1q to p0, 0, e 2t q, and the geodesic segment connecting them has length 2t. On the other hand, it has length ||σ 3 ||¨t, so ||σ 3 || " 2. So the multiplier 2 in the above definition of C guarantees that Clifford multiplication is isometric.
Remark B.2. The above Clifford multiplication is compatible with the orientation σ 1^σ2^σ3 in the sense that Cpσ 1 qCpσ 2 qCpσ 3 q " 8 is positive.
The hyperbolic metric on p 0 defines a K-equivariant isomorphism ι : p _ 0 Ñ p 0 . We define C 1 to be the composition of C with ι: We will refer to the associated map on sections as Clifford multiplication.
Definition B.2. The Dirac operator D on ΓpF S q is defined to be the composition Let X 1 , X 2 , X 3 be an oriented orthonormal basis for p 0 . The canonical isomorphism Hompp 0 , Sq » p _ 0 b S can be expressed as where X i is any orthonormal basis for p. By the discussion from §B.1.2 and §B.2.1, the Dirac operator is given by the formula where m X i denotes matrix multiplication by X i P p 0 " isup2q.
In the dual picture, D " C 1˝ΘS . Unravelling the formula for Θ S from (33), we find that D equals B.2.3. Representation-by-representation analysis of the Dirac operator. For every irreducible representation π of G, we define the π-isotypic Dirac operators ΓzGq is a subrepresentation. By the commutativity of the diagrams (34) and (35), we obtain the commutative diagram (37) where the vertical maps ι in both diagrams are induced by the inclusion π Ă L 2 p r ΓzGq.
We will show in Proposition B.5 that if π is an irreducible (unitary) representation of G and Hom K pS _ , π 8 q is non-zero, then it is 1-dimensional. For all such representations π, D π thus necessarily acts on Hom K pS _ , π 8 q by some scalar λ Dirac,π (which will be later computed in Proposition B.7). Thus, ι pHom K pS _ , π 8 qq corresponds to an eigenline for the Dirac operator D with eigenvalue λ Dirac,π . In particular, the natural decomposition which is the analogue of the Matsushima decomposition we used in [28], corresponds to the eigenspace decomposition of the Dirac operator.
B.3. Automorphic incarnation of q-form bundles and the˚d operator. Reprise all notation from §B.2. As in [28], in this setup it is more convenient to consider the operators on complex valued forms; to this effect, we will consider the K-representation p " p 0 b C. In particular, the vector bundle F^q p _ is naturally identified with the vector bundle of (complexvalued) q-forms over r ΓzH 3 . be the corresponding operator in the dual picture, described in §B.1.3. Composing with the wedge product map,^: p _ b^qp _ Ñ^q`1p _ , which is K-equivariant, recovers the exterior derivative: In the dual picture, applying formula (33) for V "^qp _ yields the following formula for d "^˝Θ^q See [8,Chapter 1,§1]. Implicitly in (38), we have used the canonical duality between^qp _ and^qp.

B.3.3.
Representation-by-representation analysis of˚d. For every irreducible representation π of G, we define the π-isotypic˚d operators p˚dq π :"˚π˝˚^π˝Θ^1 ΓzGq is a subrepresentation. By the commutativity of the diagrams (34) and (35), it follows that the diagram (39) where the vertical maps ι in both diagrams are induced by the inclusion π Ă L 2 p r ΓzGq.
Similarly to the case of the Dirac operator, we will show in Proposition B.6 that if π is an irreducible (unitary) representation of G and Hom K p^1p, π 8 q is non-zero, then it is 1dimensional. For all such representations π, p˚dq π thus necessarily acts on Hom K p^1p, π 8 q by some scalar λ˚d ,π (which will be computed in Proposition B.8). Thus, ι`Hom K p^1p, π 8 qȋ s an eigenline for the odd signature operator with eigenvalue λ˚d ,π . In particular, the natural decomposition which is the essentially the Matsushima decomposition we used in [28], corresponds to the eigenspace decomposition of odd signature operator. Notice that of course closed forms are in the kernel of˚d; as in [28], we will identify exactly which representations correspond to coexact forms.
B.4. Frobenius reciprocity and K-isotypic vectors for induced representations. Let M " K X B be the subgroup of diagonal unitary matrices. Let χ n 1 ,n 2 denote the character of M given by which is obtained by restricting the family of characters χ s 1 ,s 2 ,n 1 ,n 2 of B. Suppose π Ă L 2 p r ΓzGq is a subrepresentation isomorphic to π s 1 ,s 2 ,n 1 ,n 2 . The following Lemma will allow us to characterize whether π contributes to the spinor (resp.˚d) eigenvalue spectrum. And if π contributes, it will allow us to write down explicit vectors in Hom K pS _ , π 8 q (resp. Hom K pS _ , π 8 q) whose images in Hom K pS _ , C 8 p r ΓzGqq (resp. in Hom K pS _ , C 8 p r ΓzGqq) are Dirac eigenspinors (resp.˚d eigen 1-forms), per the discussion from §B.2 (resp. §B.3).
In particular, this reduces the problem of determining V _ -isotypic vectors to the (essentially trivial) problem of understanding of V _ as a representation of M .
B.4.1. Representations contributing to the spinor spectrum. We need to apply Lemma B.4 to the representation S _ , where again S " C 2 is the standard representation of U 2 .
Proof. We easily check that Hom K pS _ , π 8 q " 0 if π is isomorphic to some power of the determinant. So we focus on the case ππ s 1 ,s 2 ,n 1 ,n 2 .
This proves the Lemma. B.4.2. Representations contributing to the coclosed 1-form spectrum. We need to apply Lemma B.4 to the representation^1p " p, where again p " p 0 b C and p 0 " isup2q is acted on by K via the adjoint representation. Denoting again the Pauli matrices in (36) by σ i , we have that are the M -weight vectors for^1p of respective weights p1,´1q, p0, 0q and p´1, 1q. We denote the corresponding dual basis by w _ 1 , w _ 0 , w _ 1 .
This proves the Lemma.
B.5. Representation-by-representation calculation of Dirac and˚d eigenvalues. We conclude the proofs on the main results of this Appendix by computing the eigenvalues λ˚d ,π and λ Dirac,π of the eigenforms/eigenspinors corresponding to isotypic vectors.
Remark B.3. While in [28,Appendix B] we performed the computations in the natural Killing metric, and then normalized the result to obtain the answer for the hyperbolic metric, we will now work directly with the hyperbolic metric.
Summing these contributions,˚d acts on Hom K p^1p, π s 1 ,s 2 ,1,´1 q by the scalar c "˚dT pw 1 qp1q nd the proof is concluded.
Appendix C. Proof of the explicit trace formulas We will now prove the results in Section 3, starting with the more involved case of spinors in Theorem 3.3.
C.1. Isolating the spinor spectrum. We discuss suitable choices of test function to specialize the general trace formula for G in Proposition A.2. According to Proposition B.5, the irreducible unitary representations of G contributing to the spinor spectrum are precisely those isomorphic to π s 1 ,s 2 ,´1,0 or π s 1 ,s 2 ,0,´1 for purely imaginary s 1 , s 2 . To isolate the latter representations, we use the test functions of the form # H`puq`e iα`eiβ˘H`e ven H´puq`e iα´eiβ˘H´o dd We now discuss how each term in the trace formula simplifies when choosing test functions of the latter shapes.
C.1.1. Regular geometric terms for spinors. For our normalization of Haar measure (29), the covolume of the centralizer of γ equals pγ 0 q. If its conjugacy class in G is represented bŷ e {2 e iθ e iφ 0 0 e´ {2 e´i θ e iφ˙, then its complex length equals `i2θ and its corresponding summand on the geometric side of the trace formula from Proposition A.2 equals " 0¨1 |1´e `i2θ |¨|1´e´ ´i2θ |¨H˘p q¨´e ipθ`φq˘eip´θ`φq¯.
In our case, we will be interested in lifts of torsion-free lattices Γ Ă PSL 2 pCq, so that we will only need to consider the case z " 1, corresponding to φ " 0.
. formula from (42)  where the LHS is summed over all isomorphism classes of irreducible unitary representations of G isomorphic to some π s 1 ,s 2 ,´1,0 . Since H´is odd, its Fourier transform is purely imaginary. So looking imaginary parts: By (B.7), the Dirac eigenvalue on Hom K pS _ , π s 1 ,s 2 ,´1,0 q, with orientation σ 1^σ2^σ3 , equals i`s 1´s2 2˘" r 2´r1 2 . Thus by Proposition B.5, we can reexpress the above trace formula from (43) geometrically: This is the odd trace formula in Theorem 3.3; let us point out again that in (44), the Dirac operator is taken relative to the orientation σ 1^σ2^σ3 on H 3 . C.3. The trace formula for coexact 1-forms. Specializing the trace formula to isolate coexact 1-forms is more straightforward than specializing the trace formula to isolate spinors as in §C.2. We content ourselves with highlighting the main differences between specializing to 1-forms versus specializing to spinors.
‚ Irreducible unitary subrepresentations of L 2 p r ΓzGq contributing to the coclosed 1-form spectrum are precisely those isomorphic to π s 1 ,s 2 ,1,´1 and π s 1 ,s 2 ,´1,1 for s 1 , s 2 P iR. We isolate those representations using the test functions H˘puq¨`e ipα´βq˘eipβ´αq˘, where H`is even and H´is odd.
‚ The contribution of det k to the trace formula for 1-forms is non-trivial only if k " 0.
If k " 0, the contribution equals 0 for test functions H´puq¨`e ipα´βq´eipβ´αq˘f or odd H´and equals´y H`p0q for the test function H`puq¨´e ipα´βq`eipβ´αq¯" 2H`p2vq cosp2θq with H`even.
‚ In the even case for the test function 2H`p2vq cosp2θq the identity contribution is 1 2π¨v olpY q¨2pH`p0q´H 2 p0qq. From this, one readily obtains the formulas for coexact 1-forms in Theorem 3.2.

Appendix D. Admissibility of Gaussian functions
Gaussian functions were convenient to use at several points in our arguments. The next result proves that functions of sufficiently fast decay, e.g. Gaussians, are admissible for use in the trace formula. Then both the geometric and spectral sides of the spinor trace formula for closed hyperbolic 3manifolds converge absolutely for the test function f and they are equal. The same statement holds for the 1-form trace formula applied to even or odd test functions and the 0-form trace formula applied to even test functions f.
Proof. We focus on the spinor case (arguments in the other cases being identical). Convergence of the spectral side of the trace formula for f follows because p f is Schwartz and the number of spectral parameters in rν, ν`1s is quadratic in ν (see Section 5) Convergence of the geometric side of the trace formula follows by our hypothesis. Indeed, Above, we have used that ř pγqPrL,L`1s pγ 0 q has order of magnitude e 2L (cf. the discussion in Section 9) and that |1´e C pγq |¨|1´e´C pγq | has order of magnitude e pγq for pγq large.
The trace formula for f will be a consequence of the limit of the trace formulas applied to the test functions f pxq¨bpx{Rq, where b is a smooth even, compactly supported bump function with p b everywhere positive and bp0q " 1. As R Ñ 8, the geometric and spectral sides of the trace formula for f pxq¨bpx{Rq respectively converge pointwise to the geometric and spectral sides of the trace formula for f ; this is immediate for the geometric side and it follows on the spectral side because f pxq¨bpx{Rq has Fourier transform´p f˚g R¯p tq, where g R ptq :" 1 2π¨R¨p bpRtq is an approximate identity (i.e. it is everywhere positive of total mass 1 and concentrated increasingly near the origin as R Ñ 8). Here we use that for our definition of Fourier transform y f¨g " 1 2π p f˚p g. It therefore suffices to prove that the tails of the geometric and spectral sides of the trace formula, applied to the test function f pxq¨bpx{Rq, approach 0 uniformly as R Ñ 8.
Uniform smallness of the tail on the geometric side follows by our hypothesis. Indeed, ÿ pγqěkˇ pγ 0 q¨p cos or sinqθ cos φ |1´e C pγq |¨|1´e´C pγq |¨f p pγqqˇˇˇď which converges uniformly to 0 as k Ñ 8, being the tail of one fixed convergent sum (independent of R).

Department of Mathematics, Columbia University
Email address: flin@math.columbia.edu

Department of Mathematics and Statistics, McGill University
Email address: michael.lipnowski@mcgill.ca