# Closed geodesics and Frøyshov invariants of hyperbolic three-manifolds

Francesco Lin  Department of Mathematics, Columbia University
<EMAIL_ADDRESS>and  Michael Lipnowski  Department of Mathematics and
Statistics, McGill University<EMAIL_ADDRESS>
###### Abstract.

Frøyshov 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 Frøyshov 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 Frøyshov invariants of minimal
hyperbolic $L$-spaces purely in terms of volume and injectivity radius.
Second, we describe an algorithm to compute Frøyshov invariants of minimal
$L$-spaces in terms of data arising from hyperbolic geometry. As a concrete
example of our method, we compute the Frøyshov invariants for all spinc
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.

###### Contents

  1. 1 Background on Frøyshov and eta invariants
  2. 2 Torsion spinc structures and spinc length spectra
  3. 3 Trace formulas for functions, forms and spinors
  4. 4 Analytic continuation of the eta function via the odd trace formula
  5. 5 Effective local Weyl laws
  6. 6 Geometric bounds for the Frøyshov invariant
  7. 7 An explicit example: the Weeks manifold
  8. 8 The linking form of the Seifert-Weber dodecahedral space
  9. 9 Effective estimates for closed geodesics on the Seifert-Weber dodecahedral space
  10. 10 The Frøyshov invariants of the Seifert-Weber dodecahedral space
  11. A Generalities on the group $G$ and the trace formula
  12. B The odd signature and Dirac operators via representation theory
  13. C Proof of the explicit trace formulas
  14. D Admissibility of Gaussian functions

## Introduction

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
$h(Y,\mathfrak{s})\in\mathbb{Q}$ indexed by spinc 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
$d(Y,\mathfrak{s})=-2h(Y,\mathfrak{s})$ 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 $V,\varepsilon>0$, the set of
hyperbolic rational homology spheres $Y$ for which $\mathrm{vol}(Y)<V$ and
$\mathrm{inj}(Y)>\varepsilon$ is finite ([5], Chapter $5$), and therefore so
is the set of possible Frøyshov invariants $h(Y,\mathfrak{s})$.

######  Question 1.

Given $V,\varepsilon>0$, can one provide an effective upper bound on
$|h(Y,\mathfrak{s})|$ for all hyperbolic rational homology spheres with
$\mathrm{vol}(Y)<V$ and $\mathrm{inj}(Y)>\varepsilon$?

Of course, this is a challenging question even when restricting to the smaller
class of $L$-spaces with $\mathrm{vol}(Y)<V$ and
$\mathrm{inj}(Y)>\varepsilon$. Another natural question in this spirit is the
following.

######  Question 2.

Can one explicitly determine $h(Y,\mathfrak{s})$ 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 $(Y,g_{\mathrm{hyp}})$
equipped with a hyperbolic metric $g_{\mathrm{hyp}}$ 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 $\lambda_{1}^{*}>2$ (where $\lambda_{1}^{*}$ is
the first eigenvalue of the Hodge Laplacian acting on coexact $1$-forms) is a
minimal hyperbolic $L$-space; furthermore, the inequality $\lambda_{1}^{*}>2$
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 $\mathrm{vol}(Y)<V$ and
$\mathrm{inj}(Y)>\varepsilon$, then there exists an effectively computable
constant $K_{V,\varepsilon}$ for which $\lvert h(Y,\mathfrak{s})|\leq
K_{V,\varepsilon}$ for all spinc structures on $Y$. For example, if $Y$ is a
minimal hyperbolic $L$-space with $\mathrm{vol}(Y)<6.5$ and
$\mathrm{inj}(Y)>0.15$ (e.g. any rational homology sphere in the Hodgson-Weeks
census111The Hodgson-Weeks census [20] consists of approximatively $11$
thousand closed oriented hyperbolic manifolds with $\mathrm{vol}(Y)<6.5$ and
$\mathrm{inj}(Y)>0.15,$ and most of these are rational homology spheres. It is
currently not known what percentage of the such manifolds it encompasses. with
$\lambda_{1}^{*}>2$), then for every spinc structure $\mathfrak{s}$, the
inequality $\lvert h(Y,\mathfrak{s})\lvert\leq 67658$ holds.

In general, the dependence of $K_{V,\varepsilon}$ on $V$ and $\varepsilon$ 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 $<6.5$ and
injectivity radius $>0.15$, 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 $h(Y,\mathfrak{s})$ can
be expressed in terms of the eta invariants $\eta_{\mathrm{sign}}$ and
$\eta_{\mathrm{Dir}}$ of the odd signature operator $\ast d$ 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 $\eta_{\mathrm{sign}}$ and $\eta_{\mathrm{Dir}}$ in
terms of spectral and geometric data:

  * •

in the case of $\eta_{\mathrm{sign}}$, 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 $\Delta$
acting on coexact $1$-forms, and the trace formula involved spectral
parameters $t_{j}$ with $t_{j}^{2}=\lambda_{j}^{*}$. 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 $\Delta$ is the
square of $\ast d$.

  * •

in the case of $\eta_{\mathrm{Dir}}$, the relevant new geometric data is
encoded in the spinc length spectrum of $(Y,\mathfrak{s})$; 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

$G=\left\\{g\in\mathrm{GL}_{2}(\mathbb{C})\text{ for which
}|\mathrm{det}(g)|=1\right\\}.$

This should be thought as the spinc analogue of the group
$\mathrm{PGL}_{2}(\mathbb{C})=\mathrm{Isom}^{+}(\mathbb{H}^{3})$ 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) $\ast d$ 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
$<6.5$ and injectivity radius $>0.15$, the explicit inequalities

$\displaystyle\lvert\eta_{\mathrm{sign}}\lvert$ $\displaystyle\leq 108267$
$\displaystyle\lvert\eta_{\mathrm{Dir}}\lvert$ $\displaystyle\leq 108249$

hold, where the second estimate is independent of the choice of spinc
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
$[-2\cdot\mathrm{inj}(Y),2\cdot\mathrm{inj}(Y)]$, 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 $\mathsf{SW}$. 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 $\mathsf{SW}$
is not Haken [11]. In [29] we used our techniques to show that $\mathsf{SW}$
is a minimal hyperbolic $L$-space by taking into account its large symmetry
group. The next result determines its Frøyshov invariants for all the spinc
structures on $\mathsf{SW}$ (recall that
$H_{1}(\mathsf{SW};\mathbb{Z})=(\mathbb{Z}/5\mathbb{Z})^{3})$.

######  Theorem 0.2.

Let $\mathfrak{s}_{0}$ be the unique spin structure on $\mathsf{SW}$, and
consider a spinc structure $\mathfrak{s}=\mathfrak{s}_{0}+x$ on $\mathsf{SW}$
for $x\in H^{2}(\mathsf{SW},\mathbb{Z})$. Then the Frøyshov invariant
$h(\mathsf{SW},\mathfrak{s})$ is computed as in the following table, according
to the value of the linking form $\mathsf{lk}(x,x)\in\mathbb{Q}/\mathbb{Z}$.

$\mathsf{lk}(x,x)$ | $h(\mathsf{SW},\mathfrak{s})$
---|---
$0$ | $-1/2$
$1/5$ | $-1/10$
$2/5$ | $-7/10$
$3/5$ | $-3/10$
$4/5$ | $1/10$

######  Remark 0.2.

As we will see in the proof, the case $\mathsf{lk}(x,x)=0$ really encompasses
two distinct cases: the unique spin structure and a family of $24$ (non-spin)
spinc structures.

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 $\mathsf{SW}$. 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 $\eta_{\mathrm{sign}}$ and $\eta_{\mathrm{Dir}}$ 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 $\mathsf{SW}$ 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 $\eta_{\mathrm{sign}}$ 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
$\eta_{\mathrm{sign}}$ 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 spinc 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 $\eta_{\mathrm{Dir}}$ 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 three-manifolds 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,

$\eta_{\mathrm{Dir}}=0.989992\dots$

This relies on the fact that the Weeks manifold is a minimal hyperbolic
$L$-space, together with the computation of $\eta_{\mathrm{sign}}$ 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
$\eta_{\mathrm{Dir}}$ provided one can access the spinc length spectrum of
$(Y,\mathfrak{s})$; in turn, we will describe an algorithm to compute
$\mathrm{spin}^{c}$ length spectra taking as input information computed using
SnapPy. In particular, our method could in principle be applied to compute the
invariants $\eta_{\mathrm{Dir}}$ 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
(spinc) 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, spinc 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.

Acknowledgements. We are greatly indebted to Nathan Dunfield for all the help
with SnapPy. The first author was partially supported by NSF grant DMS-1948820
and an Alfred P. Sloan fellowship.

##  1\. Background on Frøyshov and eta invariants

In this section we review some background topics that will be central for the
purposes of the paper.

###  1.1. Formal structure of monopole Floer homology and Frøyshov invariants

In [23] the authors associate to each three-manifold $Y$ three
$\mathbb{Z}[U]$-modules fitting in a long exact sequence

$\cdots\stackrel{{\scriptstyle i_{*}}}{{\longrightarrow}}\smash{\mskip
2.0mu\raise 15.93944pt\hbox to0.0pt{\begin{turn}{180.0}
{$\widehat{\phantom{\mathit{HM}}}$}
\end{turn}\hss}\mskip-2.0mu\mathit{HM}}{\vphantom{\widehat{\mathit{HM}}}}{}_{*}(Y)\stackrel{{\scriptstyle
j_{*}}}{{\longrightarrow}}\widehat{\mathit{HM}}_{*}(Y)\stackrel{{\scriptstyle
p_{*}}}{{\longrightarrow}}\overline{\mathit{HM}}_{*}(Y)\stackrel{{\scriptstyle
i_{*}}}{{\longrightarrow}}\cdots.$

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 spinc structures on $Y$; for example we have

$\smash{\mskip 2.0mu\raise 15.93944pt\hbox to0.0pt{\begin{turn}{180.0}
{$\widehat{\phantom{\mathit{HM}}}$}
\end{turn}\hss}\mskip-2.0mu\mathit{HM}}{\vphantom{\widehat{\mathit{HM}}}}{}_{*}(Y)=\bigoplus_{\mathfrak{s}\in\mathrm{Spin}^{c}(Y)}\smash{\mskip
2.0mu\raise 15.93944pt\hbox to0.0pt{\begin{turn}{180.0}
{$\widehat{\phantom{\mathit{HM}}}$}
\end{turn}\hss}\mskip-2.0mu\mathit{HM}}{\vphantom{\widehat{\mathit{HM}}}}{}_{*}(Y,\mathfrak{s}).$

The reduced Floer homology group ${\mathit{HM}}_{*}(Y,\mathfrak{s})$ 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 spinc struture admit an absolute grading by
a $\mathbb{Z}$-coset in $\mathbb{Q}$, and the action of $U$ has degree $-2$.
Furthermore, we have that

$\overline{\mathit{HM}}_{*}(Y,\mathfrak{s})\cong\mathbb{Z}[U,U^{-1}]$

as graded modules (up to an overall shift),
$\widehat{\mathit{HM}}_{*}(Y,\mathfrak{s})$ 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
${\mathit{HM}}_{*}(Y,\mathfrak{s})=0$ for all spinc structures. Given a spinc
rational homology sphere $(Y,\mathfrak{s})$, denote the minimum degree of a
non-zero element in
$i_{*}\left(\overline{\mathit{HM}}_{*}(Y,\mathfrak{s})\right)\subset\smash{\mskip
2.0mu\raise 15.93944pt\hbox to0.0pt{\begin{turn}{180.0}
{$\widehat{\phantom{\mathit{HM}}}$}
\end{turn}\hss}\mskip-2.0mu\mathit{HM}}{\vphantom{\widehat{\mathit{HM}}}}{}_{*}(Y,\mathfrak{s})$
by $-2h(Y,\mathfrak{s})$. The quantity $h(Y,\mathfrak{s})$ is then called the
Frøyshov invariant of $(Y,\mathfrak{s})$.

###  1.2. 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 $\Omega^{\mathrm{even}}$ as

$(-1)^{p}(\ast d-d\ast)\text{ on $2p$-forms.}$

We identify $2$-forms and $1$-forms using $\ast$, so that the operator is

$\left(\begin{array}[]{cc}\ast d&d\\\ d^{*}&0\end{array}\right)$

acting on $\Omega^{1}\oplus\Omega^{0}$. 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

$\eta_{\mathrm{sign}}(s)=\sum_{\lambda\neq 0\text{ eigenvalue
}}\frac{\mathrm{sgn}(\lambda)}{|\lambda|^{s}};$

this sum defines a holomorphic function for $\mathrm{Re}(s)$ 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,
$\eta_{\mathrm{sign}}=\eta_{\mathrm{sign}}(0)$ measures the spectral asymmetry
of the operator. Similarly, the same procedure works for the Dirac operator
$D_{B_{0}}$ (and its perturbations), leading to $\eta_{\mathrm{Dir}}$.

######  Remark 1.1.

For our purposes it is convenient to notice (cf. [2, Proposition $4.20$]) that
the odd signature operator (assuming for simplicity $b_{1}(Y)=0$) under the
Hodge decomposition $\Omega^{1}=d\Omega^{0}\oplus d^{*}\Omega^{2}$ can be
written as

$\left(\begin{array}[]{ccc}\ast d&0&0\\\ 0&0&d\\\ 0&d^{*}&0\end{array}\right)$

acting on $d^{*}\Omega^{2}\oplus d\Omega^{0}\oplus\Omega^{0}$. Now, the block

$\left(\begin{array}[]{cc}0&d\\\ d^{*}&0\end{array}\right)$

has symmetric spectrum because $d$ and $d^{*}$ are adjoints; in particular we
have for $\mathrm{Re}(s)$ large enough

(1)
$\eta_{\mathrm{sign}}(s)=\sum_{t_{j}}\frac{\mathrm{sgn}(t_{j})}{|t_{j}|^{s}};$

where the sum runs only on the eigenvalues $\\{t_{j}\\}$ of $\ast d$ on
coexact $1$-forms (notice that all the $t_{j}$ are non-zero). In particular,
$\eta_{\mathrm{sign}}$ coincides with the spectral asymmetry of the action of
$\ast d$ on coexact $1$-forms.

######  Remark 1.2.

We have $(\ast d)^{2}=\Delta$ when acting on coexact $1$-forms, and therefore
the squares of the parameters $t_{j}$ are exactly the eigenvalues
$\lambda_{j}^{*}$ of $\Delta$ 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.

######  Proposition 1.1.

Suppose $(Y,g)$ is a minimal $L$-space. Then for each spinc structure
$\mathfrak{s}$,

$h(Y,\mathfrak{s})=-\frac{\eta_{\mathrm{sign}}}{8}-\frac{\eta_{\mathrm{Dir}}}{2}$

where $\eta_{\mathrm{Dir}}$ is the eta invariant of the Dirac operator
$D_{B_{0}}$ corresponding to the flat connection on the determinant line
bundle $B^{t}_{0}$.

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 $\lambda_{1}^{*}$ satisfies
$\lambda_{1}^{*}>2$ 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 $-2h(Y,\mathfrak{s})$ 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 spinc structures. As we only
consider rational homology spheres, the $\mathrm{spin}^{c}$ structures in our
context are automatically torsion. Consider a cobordism $(W,\mathfrak{s}_{W})$
from $S^{3}$ to $(Y,\mathfrak{s})$; equip $S^{3}$ with a round metric and a
small admissible perturbation, and denote by $[\mathfrak{a}_{0}]$ 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 $[\mathfrak{a}]$ of $Y$
the rational number

(2)
$\mathrm{gr}^{\mathbb{Q}}([\mathfrak{a}])=-\mathrm{gr}([\mathfrak{a}_{0}],W^{*},\mathfrak{s}_{W},[\mathfrak{a}])+\frac{c_{1}(\mathfrak{s}_{W})^{2}-2\chi(W)-3\sigma(W)}{4}$

where:

  * •

$\mathrm{gr}([\mathfrak{a}_{0}],W^{*},\mathfrak{s}_{W},[\mathfrak{a}])$ is the
expected dimension of the moduli space of solutions of Seiberg-Witten
equations in the spinc structure $\mathfrak{s}_{W}$ that converge to
$[\mathfrak{a}_{0}]$ and $[\mathfrak{a}]$. Concretely, this is the index of
the linearized equations, after gauge fixing.

  * •

$c_{1}(\mathfrak{s}_{W})^{2}\in\mathbb{Q}$ is the self-intersection of the
class $c_{1}(\mathfrak{s}_{W})\in H^{2}(W;\mathbb{Z})$. Recall that this is
defined as

$(\tilde{c}\cup\tilde{c})[W,\partial W]$

where $\tilde{c}\in H^{2}(W,\partial W,\mathbb{Q})$ is any class whose image
in $H^{2}(W,\mathbb{Q})$ is the same as the image $c_{1}(\mathfrak{s}_{W})$
under the change of coefficient map $H^{2}(W,\mathbb{Z})\rightarrow
H^{2}(W,\mathbb{Q})$.

For our purposes, it will be convenient to work with a closed manifold $X$
with boundary $Y$ over which $\mathfrak{s}$ 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)
$\mathrm{gr}^{\mathbb{Q}}([\mathfrak{b}])=-\mathrm{gr}(X^{*},\mathfrak{s}_{X},[\mathfrak{b}])+\frac{c_{1}(\mathfrak{s}_{X})^{2}-2\chi(X)-3\sigma(X)-2}{4}\in\mathbb{Q}$

holds, where $\mathrm{gr}(X^{*},\mathfrak{s}_{X},[\mathfrak{b}])$ 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 $(Y,g)$ is a minimal $L$-space. Then for each spinc structure
$\mathfrak{s}$, the Dirac operator $D_{B_{0}}$ corresponding to the flat
connection $B_{0}^{t}$ 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

$\mathcal{L}_{\delta}=\mathcal{L}+\frac{\delta}{2}\|\Psi\|^{2}_{L^{2}},$

for which the corresponding perturbed Dirac operator at $(B_{0},0)$ is
$D_{B_{0}}+\delta$. For small values of $\pm\delta$, 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 $-2h(Y,\mathfrak{s})$. This implies that the moduli
space of solutions to the perturbed equations connecting them on a product
cobordism $\mathbb{R}\times Y$ (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 $D_{B_{0}}+f(t)$, where $f(t)$ is a monotone function
with $f(t)=\pm\delta$ for $t>1$ and $t<-1$ 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 $[\mathfrak{b}_{0}]$, which lies over
the reducible solution $[B_{0},0]$. Under our assumptions, its absolute
grading is exactly $-2h(Y,\mathfrak{s})$. Taking $A$ to be a spinc connection
of $\mathfrak{s}_{X}$ restricting to $B_{0}$ in a neighborhood of the
boundary, linearizing the equations at $(A,0)$ we have

$\mathrm{gr}(X^{*},\mathfrak{s}_{X},[\mathfrak{b}_{0}])=\mathrm{ind}_{L^{2}(X)}(d^{*}+d^{+})+2\mathrm{ind}^{\mathbb{C}}_{L^{2}(X)}(D^{+}_{A})$

where

$d^{*}+d^{+}:i\Omega^{1}(X)\rightarrow i\Omega^{0}(X)\oplus i\Omega^{+}(X)$

has index

${b_{1}(X)-b_{2}^{+}(X)-1}=-\frac{\chi(X)+\sigma(X)+1}{2}.$

The Atiyah-Patodi-Singer for the odd signature operator says in our case

$\sigma(X)=\eta_{\mathrm{sign}}+\frac{1}{3}\int_{X}p_{1}$

where $p_{1}$ is the first Pontryagin form of the metric, and
$\eta_{\mathrm{sign}}$ 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

$\mathrm{ind}^{\mathbb{C}}_{L^{2}(X)}(D^{+}_{A})=\frac{1}{2}\eta_{D_{\mathrm{Dir}}}+\frac{1}{8}\int_{X}\left(-\frac{1}{3}p_{1}+c_{1}(A)^{2}\right)$

where $c_{1}(A)$ 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

$\mathrm{gr}^{\mathbb{Q}}([\mathfrak{b}_{0}])=\frac{1}{4}\eta_{\mathrm{sign}}+\eta_{\mathrm{Dir}},$

and the result follows.

##  2\. Torsion spinc structures and spinc length spectra

Recall that a closed geodesic $\gamma$ in an oriented hyperbolic three-
manifold admits complex length

$\mathbb{C}\ell(\gamma):=\ell(\gamma)+i\cdot\mathrm{hol}(\gamma)\in\mathbb{R}^{>0}+i\mathbb{R}/2\pi\mathbb{Z}.$

In this section we discuss a refinement of this notion when the manifold is
equipped with a torsion spinc 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 spinc structures and $G$

Let us begin by discussing with the simpler case of a genuine spin structure
$\mathfrak{s}_{0}$ on $Y$. The choice of a spin structure $\mathfrak{s}_{0}$
allows us to lift the holonomy of a closed geodesic $\gamma:S^{1}\rightarrow
Y$ from an element in $\mathbb{R}/2\pi\mathbb{Z}$ to an element in
$\mathbb{R}/4\pi\mathbb{Z}$. From a topological perspective, this is because,
using the trivial spin structure on the tangent bundle of $\gamma$,
$\mathfrak{s}_{0}$ this induces a spin structure on the normal bundle of
$\gamma$; 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
$\mathrm{hol}(\gamma)/2$ as an element in $\mathbb{R}/2\pi\mathbb{Z}$.

######  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
$\mathrm{hol}(\gamma^{n})/2=n\cdot(\mathrm{hol}(\gamma)/2)$ in
$\mathbb{R}/2\pi\mathbb{Z}$. This does not hold if we choose the Lie spin
structure.

From the point of view of hyperbolic geometry, thinking of
$\pi_{1}(Y)\subset\mathrm{Isom}^{+}(\mathbb{H}^{3})\cong\mathrm{PSL}_{2}(\mathbb{C})$,
a spin structure is a lift

${\mathrm{SL}_{2}(\mathbb{C})}$${\pi_{1}(Y)}$${\mathrm{PSL}_{2}(\mathbb{C})}$

where the vertical map is the quotient by the subgroup $\\{\pm 1\\}$. Notice
that all orientable $3$-manifolds 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
$\pi_{1}(Y)\rightarrow\mathbb{\\{}\pm 1\\}$, i.e. an element in
$H^{1}(Y,\mathbb{Z}/2)$. From this viewpoint, recall that complex length of a
hyperbolic element $\gamma$ is given by

$\mathbb{C}\ell(\gamma)=2\log\lambda_{\gamma}$

where $\lambda_{\gamma}$ is the root of

$x^{2}-(\mathrm{tr}\gamma)x+1=0$

with $|\lambda_{\gamma}|>1$. Of course, for an element
$\gamma\in\mathrm{PSL}_{2}(\mathbb{C})$ the trace is well-defined only up to
sign, and we see that the two choices of the trace correspond to
$\pm\lambda_{\gamma}$; this implies that the argument of
$\lambda_{\gamma}^{2}$, which is $\mathrm{hol}(\gamma)$, is well defined
modulo $2\pi$.

It is then clear that a spin structure, which provides a well-defined trace,
also fixes a choice between $\pm\lambda_{\gamma}$; the argument of this
choice, which is $\mathrm{hol}(\gamma)/2,$ is well defined modulo $2\pi$. Very
concretely, this is implies that the lift
$\tilde{\gamma}\in\mathrm{SL}_{2}(\mathbb{C})$ is conjugate to

$\left(\begin{array}[]{cc}e^{\frac{\ell(\gamma)}{2}+i\frac{\mathrm{hol}(\gamma)}{2}}&0\\\
0&e^{-\frac{\ell(\gamma)}{2}-i\frac{\mathrm{hol}(\gamma)}{2}}\end{array}\right).$

The case of general torsion spinc structures is analogous, where we use
instead the group

$G=\\{g\in\mathrm{GL}_{2}(\mathbb{C}):|\det(g)|=1\\}\subset\mathrm{GL}_{2}(\mathbb{C}).$

There is a natural map

$\pi:G\rightarrow\mathrm{PSL}_{2}(\mathbb{C})$

obtained by expressing

$G=\left(\mathrm{SL}_{2}(\mathbb{C})\times U_{1}\right)/\\{\pm 1\\},$

where $U_{1}\subset G$ is the subgroup of multiples of the identity, and
projecting onto the first factor. A torsion spinc structure, together with a
flat spinc connection, is then a homomorphism

$\varphi:\pi_{1}(Y)\rightarrow G$

for which $\pi\circ\varphi:\pi_{1}(Y)\rightarrow\mathrm{PSL}_{2}(\mathbb{C})$
is the inclusion map. We see that two spinc structures then differ by a
homomorphism

$\pi_{1}(Y)\rightarrow U_{1},$

which can be thought of as an element in $H^{1}(Y,\mathbb{R}/\mathbb{Z})$. The
Bockstein long exact sequence for the coefficients

$0\rightarrow\mathbb{Z}\rightarrow\mathbb{R}\rightarrow\mathbb{R}/\mathbb{Z}\rightarrow
0$

reads in this case

$H^{1}(Y;\mathbb{Z})\rightarrow H^{1}(Y,\mathbb{R})\rightarrow
H^{1}(Y,\mathbb{R}/\mathbb{Z})\rightarrow H^{2}(Y,\mathbb{Z})\rightarrow
H^{2}(Y,\mathbb{R}).$

The latter allows us to interpret topological classes of torsion spinc
structure as an affine space over the torsion of $H^{2}(Y;\mathbb{Z})$.

Throughout the paper, we will study rational homology spheres, for which all
spinc structures are torsion. In this case, we have identifications

$H^{1}(Y,\mathbb{Q}/\mathbb{Z})\cong H^{1}(Y,\mathbb{R}/\mathbb{Z})\cong
H^{2}(Y;\mathbb{Z})$

We will further fix a base spin structure $\mathfrak{s}_{0}$ and accordingly
identify spinc structures with $H^{1}(Y,\mathbb{Q}/\mathbb{Z})$. For
simplicity, $\varphi$ will interchangeably refer to both the lift and the
difference homomorphism. In this case, the element $\gamma$ has lift conjugate
to

$\left(\begin{array}[]{cc}e^{i\varphi(\gamma)}e^{\frac{\ell(\gamma)}{2}+i\frac{\mathrm{hol}(\gamma)}{2}}&0\\\
0&e^{i\varphi(\gamma)}e^{-\frac{\ell(\gamma)}{2}-i\frac{\mathrm{hol}(\gamma)}{2}}\end{array}\right)$

and the spinc length spectrum keeps track of the value $\varphi(\gamma)$. We
will refer to the latter as the twisting character.

###  2.2. Computations of spinc lifts

For eventual use of the trace formula for spinors on $G,$ we need to
explicitly compute the torsion spinc data, namely
$\frac{\mathrm{hol}(\gamma)}{2}$ and $\phi(\gamma)$ for representatives of all
conjugacy classes in $\Gamma$ up to some specified real length. We refer to
this data as the _spin c length spectrum_ of $(Y,\mathfrak{s})$. We discuss a
concrete algorithm to compute such information, taking as input data provided
by SnapPy.

####  2.2.1. Input for spinc length spectrum: SnapPy Dirichlet domain and
complex length spectrum

In order to compute the spinc 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 $\Gamma=\pi_{1}(Y)$ centered at $o=(1,0,0,0)$ in
the hyperboloid model of $\mathbb{H}^{3},$ i.e. the upper sheet of

$\bigg{\\{}(t,x,y,z)\text{ such that }Q(t,x,y,z)=-1\bigg{\\}},$

where $Q(t,x,y,z)=-t^{2}+x^{2}+y^{2}+z^{2}$. In particular, this includes a
list of all elements $\gamma$ of the identity component of the orthogonal
group $\mathrm{SO}(Q)^{0}$ for which some domain within the bisector of $o$
and $\gamma o$ is a face $F\subset D.$ For hyperbolic 3-manifolds $Y,$ such
elements $\gamma$ come in inverse pairs: for the above element
$\gamma,\gamma^{-1}F$ is the face of $D$ opposite $F.$ The group $\Gamma$ is
generated by the face-pairing elements (see §2.2.4).

  * •

For each conjugacy class $C$ in $\Gamma$ with $\ell(C)\leq R$, a matrix
$g_{C}$ in $\mathrm{SO}(Q)^{0},$ which is the image in $\mathrm{SO}(Q)^{0}$ of
some element of $\Gamma$ representing $C.$ One can extract the complex length
of $C$ from $g_{C}$, but of course it contains more information.

####  2.2.2. Converting $\Gamma\subset\mathrm{SO}(Q)^{0}$ to
$\Gamma\subset\mathrm{PSL}_{2}(\mathbb{C})$

Minkowski space $\mathbb{R}^{1,3}$ can be identified with the space of
$2\times 2$ hermitian matrices $i\mathfrak{u}_{2}$ via the map

$(t,x,y,z)\mapsto\left(\begin{array}[]{cc}t+z&x-iy\\\
x+iy&t-z\end{array}\right).$

Under this identification, the quadratic form $Q$ equals the determinant of
the above matrix. The group $\mathrm{PSL}_{2}(\mathbb{C})$ acts on
$i\mathfrak{u}_{2}$ by

$g\cdot X=gXg^{\ast},$

preserving the determinant. The resulting homomorphism

$\mathrm{PSL}_{2}(\mathbb{C})\xrightarrow{\sim}\mathrm{SO}(Q)^{0}$

is in fact an isomorphism. To transform $\Gamma$ to a corresponding subgroup
of $\mathrm{PSL}_{2}(\mathbb{C})$ amounts to inverting the above isomorphism.
In turn, this amounts to solving the following linear algebra problem:

Given $f\in\mathrm{End}(i\mathfrak{u}_{2})$ preserving $Q,$ find
$g\in\mathrm{GL}_{2}(\mathbb{C})$ satisfying

(4) $Xg^{\ast}=\left(\det(g)\cdot g^{-1}\right)f(X)\text{ for all }X\in
i\mathfrak{u}_{2}.$

The entries of $\det(g)\cdot g^{-1}$ are linear in the entries of $g,$ so (4)
defines a linear algebra problem (over $\mathbb{R}$), which is readily solved.
Rescaling the solution $g$ to (4) as needed, we solve (4) with some matrix
$g\in\mathrm{PSL}_{2}(\mathbb{C}).$

####  2.2.3. Reduction theory via Dirichlet domains for conjugacy classes
$C\subset\Gamma$

We discuss how to express a given element $\gamma\in\Gamma$ in terms of the
face-pairing elements $\gamma_{i}$ of $D$.

######  Lemma 2.1.

Suppose $x\in\mathbb{H}^{3}\setminus D.$ There is some face pairing element
$\gamma$ for which $d(\gamma^{-1}\cdot x,o)<d(x,o).$

###### Proof.

Because $\gamma_{1},\ldots,\gamma_{m}$ support the faces of the Dirichlet
domain $D,$

$D=\\{x\in\mathbb{H}^{3}:d(x,o)\leq d(x,\gamma_{i}\cdot o)\text{ for
}i=1,\ldots,m\\}.$

In particular, if $x\notin D,$ then

$d(x,o)>d(x,\gamma_{i}\cdot o)=d(\gamma_{i}^{-1}\cdot x,o)$

for some face-pairing element $\gamma_{i}.$ ∎

This yields the following Dirichlet domain reduction algorithm for $\Gamma$:

  * •

Initialize $x=\gamma\cdot o.$ Let $L=[]$ be an empty list. Note: if $\gamma$
is not the identity element, then $x\notin D.$

  * •

While $x$ is not in $D$: find a face pairing element $\gamma$ for which
$d(\gamma^{-1}x,o)<d(x,o)$; this is always possible by Lemma 2.1. Then:

    * –

Append $\gamma$ to the right end of $L.$

    * –

Replace $x$ by $\gamma^{-1}x.$

  * •

Once $x\in D,$ output $\delta=\prod_{g\in L}g,$ product taken in left-to-right
order.

######  Lemma 2.2.

The Dirichlet domain reduction algorithm for $\Gamma$ terminates. Its output
equals $\gamma.$

###### Proof.

Let $x=\gamma\cdot o.$ If the while loop did not terminate, we would find an
infinite sequence of face pairing elements $\gamma_{1},\gamma_{2},\ldots$ for
which $x,\gamma_{1}^{-1}\cdot x,\gamma_{2}^{-1}\cdot\gamma_{1}^{-1}\cdot
x,\ldots$ all do not lie in $D$ and for which

$d(x,o)>d(\gamma_{1}^{-1}\cdot x,o)>d(\gamma_{2}^{-1}\cdot x,o)>\cdots$

But the latter is not possible since the action of $\Gamma$ on
$\mathbb{H}^{3}$ is properly discontinuous. Suppose now
$L=[\gamma_{1},\cdots,\gamma_{n}]$ at the point of the algorithm when $x\in
D.$ That means that

$\gamma_{n}^{-1}\gamma_{n-1}^{-1}\cdots\gamma_{1}^{-1}\cdot(\gamma\cdot o)\in
D.$

The latter point is evidently $\Gamma$-equivalent to $o$ and lies in $D.$
Since $o$ does not lie on the boundary of $D,$ $o$ is the unique point of $D$
which is $\Gamma$-equivalent to $o.$ So

$\left(\gamma_{n}^{-1}\gamma_{n-1}^{-1}\cdots\gamma_{1}^{-1}\cdot\gamma\right)\cdot
o=o.$

Since the action of $\Gamma$ on $\mathbb{H}^{3}$ is fixed-point free, it
follows that $\gamma=\gamma_{1}\cdots\gamma_{n}.$∎

####  2.2.4. Computing one spin structure

As mentioned above, it is convenient to fix a base genuine spin structure
$\mathfrak{s}_{0}$ (so that all the others spinc structures are obtained by
twisting it via a character). To compute the corresponding lift
$\tilde{\Gamma}\subset\mathrm{SL}_{2}(\mathbb{C})$, we use the presentation of
$\Gamma\subset\mathrm{PSL}_{2}(\mathbb{C})$ afforded to us by the Dirichlet
domain $D.$ Let $\Gamma_{\mathrm{abs}}$ denote the (abstract) group with a
generator $[\gamma]$ for each face pairing element $\gamma$ and satisfying the
following relations:

  * •

(opposite face relations) $[\gamma]\cdot[\gamma^{-1}]=1$ for all face pairing
elements $\gamma$

  * •

(edge cycle relations) The edges of $D$ are partitioned into _edge cycles_.
This is an ordered sequence of edges $e_{1},e_{2},\ldots,e_{n}$ for which
$\gamma_{1}\cdot e_{1}=e_{2},\gamma_{2}\cdot
e_{2}=e_{3},\ldots,\gamma_{n}\cdot e_{n}=e_{1}$ for some
$\gamma_{1},\gamma_{2}\dots,\gamma_{n}$. Every edge cycle yields a
corresponding relation: $[\gamma_{n}]\cdots[\gamma_{1}]=1.$

The opposite face and edge cycle relations hold for the corresponding elements
of $\Gamma.$ The assignment

$\displaystyle\iota:\mathrm{\Gamma_{\mathrm{abs}}}$
$\displaystyle\rightarrow\Gamma\subset\mathrm{PSL}_{2}(\mathbb{C})$ (5)
$\displaystyle[\gamma]$ $\displaystyle\mapsto\gamma$

thus extends to a well-defined homomorphism. According to the Poincaré
polyhedron theorem [30], $\iota$ is an isomorphism.

For every face-pairing element
$\gamma\in\Gamma\subset\mathrm{PSL}_{2}(\mathbb{C}),$ choose arbitrarily a
lift $\widetilde{\gamma}\in\mathrm{SL}_{2}(\mathbb{C}).$ Because $\iota$ is
well-defined, all of the defining relations have an associated sign. For
example, if $[\gamma_{n}]\cdots[\gamma_{1}]=1$ is an edge cycle relation, then

$\widetilde{\gamma_{n}}\cdots\widetilde{\gamma_{1}}=\epsilon_{[\gamma_{n}]\cdots[\gamma_{1}]}\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right),$

where $\epsilon_{[\gamma_{n}]\cdots[\gamma_{1}]}=\pm 1.$ Let
$\epsilon_{[\gamma]}=\pm 1$ be unknowns we will solve for. Then

$\gamma\mapsto\epsilon_{[\gamma]}\cdot\widetilde{\gamma}$

extends to a well-defined lift of $\Gamma$ to
$\widetilde{\Gamma}\subset\mathrm{SL}_{2}(\mathbb{C})$ iff the opposite face
and edge cycle relations are satisfied. This immediately reduces to the
following linear algebra problem (over $\mathbb{Z}/2\mathbb{Z}$):

$\begin{cases}\epsilon_{[\gamma]}\epsilon_{[\gamma^{-1}]}=\epsilon_{[\gamma]\cdot[\gamma^{-1}]}&\text{
opposite face relations}\\\
\epsilon_{[\gamma_{n}]}\cdots\epsilon_{[\gamma_{1}]}=\epsilon_{[\gamma_{n}]\cdots[\gamma_{1}]}&\text{
edge cycle relations}.\\\ \end{cases}$

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 $\widetilde{\Gamma}$ is a torsor for
$H^{1}(\Gamma\backslash\mathbb{H}^{3},\mathbb{Z}/2).$ No preferred lift exists
in general. We note, however, that $\mathsf{SW},$ the protagonist of the
second part of our paper, satisfies $H^{1}(\mathsf{SW},\mathbb{Z}/2)=0,$ so
$\pi_{1}(\mathsf{SW})$ admits a unique lift to
$\widetilde{\pi_{1}(\mathsf{SW})}\subset\mathrm{SL}_{2}(\mathbb{C}).$

####  2.2.5. Computing all homomorphisms
$\Gamma\rightarrow\mathbb{Q}/\mathbb{Z}$

From the previous subsection, $\Gamma_{\mathrm{abs}}$ gives a presentation for
$\Gamma.$ Let $e_{1},\cdots,e_{m}$ denote the images of the face-pairing
elements $\gamma_{1},\ldots,\gamma_{m}$ in the abelianization
$\Gamma^{\mathrm{ab}}.$ The abelian group $\Gamma^{\mathrm{ab}}$ can be
presented as

$\mathbb{Z}\langle e_{1},e_{2},\dots,e_{m}\rangle/R,$

where $R$ are the abelian versions of the opposite face and edge cycle
relations from Subsection 2.2.4. We assume that $b_{1}(Y)=0$, so that
$\Gamma^{\mathrm{ab}}=H_{1}(Y,\mathbb{Z})$ is a finite group. Using the Smith
normal form, we compute a basis $b_{1},\dots,b_{m}$ of $\mathbb{Z}\langle
e_{1},e_{2},\dots,e_{m}\rangle$ for which a basis for the span of $R$ is given
$d_{1}b_{1},\dots,d_{m}b_{m}$ for some non-zero integers
$d_{1},d_{2},\dots,d_{m}$ with $d_{i}\lvert d_{i+1}$. In particular,

$\Gamma^{\mathrm{ab}}=H_{1}(Y,\mathbb{Z})=\bigoplus\mathbb{Z}/d_{i}\mathbb{Z}.$

Homomorphisms from $\Gamma^{\mathrm{ab}}$ to $\mathbb{Q}/\mathbb{Z}$ are
uniquely determined by the images $\mathbf{y}:=(y_{1},\ldots,y_{m})$ of
$b_{1},\ldots,b_{m}$; we define $\phi_{\mathbf{y}}$ as the one sending

$b_{i}\mapsto y_{i}\in\left(\frac{1}{d_{i}}\mathbb{Z}\right)/\mathbb{Z}.$

If $A=(b_{1}|\cdots|b_{m})$ is the matrix with $j$th column $b_{j},$ then

$\mathbf{y}A^{-1}=(\phi_{\mathbf{y}}(e_{1}),\ldots,\phi_{\mathbf{y}}(e_{m})).$

####  2.2.6. Indexing all lifts of $\Gamma$ to $\widetilde{\Gamma}\subset G$

We index the lifts of $\Gamma\subset\mathrm{PSL}_{2}(\mathbb{C})$ to
$\widetilde{\Gamma}\subset G$ as follows:

  * •

Compute one spin lift
$s_{0}:\Gamma\rightarrow\Gamma_{\mathrm{spin}}\subset\mathrm{SL}_{2}(\mathbb{C}),$
by the procedure described in §2.2.4.

  * •

A general lift of $\Gamma$ to $G$ is of the form
$\gamma\mapsto\phi(\gamma)\cdot s_{0}(\gamma)$ for some twisting character
$\phi:\Gamma\rightarrow\mathrm{U}_{1}.$

Thus,

$\left\\{\gamma\mapsto e^{2\pi i\phi_{\mathbf{y}}(\gamma)}\cdot
s(\gamma):y_{1}\in\left(\frac{1}{d_{1}}\mathbb{Z}\right)/\mathbb{Z},\ldots,y_{m}\in\left(\frac{1}{d_{m}}\mathbb{Z}\right)/\mathbb{Z}\right\\}$

parametrizes all lifts of $\Gamma$ to $G.$

####  2.2.7. Computing the spinc length spectrum

Suppose we have computed one lift
$s_{0}:\Gamma\rightarrow\mathrm{SL}_{2}(\mathbb{C})$ specified by the images
of the face-pairing generators for $D$; SnapPy provides the images in
$\mathrm{SO}(Q)^{0}$ 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 $\phi:\Gamma\rightarrow\mathbb{Q}/\mathbb{Z},$
specified by its values on the face-pairing generators, e.g. the homomorphisms
$\phi_{\mathbf{y}}$ from §2.2.5.

For every $\Gamma$-conjugacy class of translation length at most $R,$ SnapPy
specifies the image in $\mathrm{SO}(Q)^{0}$ of some representative element
$\gamma$ of its conjugacy class. Notice that $\gamma$ 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
$\gamma=\gamma_{1}\cdots\gamma_{k}$ for some face-pairing elements
$\gamma_{1},\ldots,\gamma_{k}$ for $D.$ Having computed $s_{0}$ on the face-
pairing generators, $s_{0}(\gamma)=s_{0}(\gamma_{1})\cdots s_{0}(\gamma_{k})$
is some explicit element of $\mathrm{SL}_{2}(\mathbb{C}).$ We readily compute
its $\mathrm{SL}_{2}(\mathbb{C})$-conjugacy class:

$s_{0}(\gamma)\sim_{\mathrm{SL}_{2}(\mathbb{C})}\left(\begin{array}[]{cc}e^{u/2}e^{i\theta}&0\\\
0&e^{-u/2}e^{-i\theta}\end{array}\right),$

and we have that $u=l(\gamma)$ and
$\theta=\mathrm{hol}(\gamma)/2\in\mathbb{R}/(2\pi\mathbb{Z})$.

Finally, since the values of the twisting character $\phi$ are known on the
face-pairing generators, we readily compute
$\phi(\gamma)=\phi(\gamma_{1})+\cdots+\phi(\gamma_{k})=:\varphi(\gamma).$ In
the lift $\widetilde{\Gamma}\subset G$ of $\Gamma$ corresponding to $s_{0}$
and $\phi$, we have

$\widetilde{\gamma}\sim_{G}e^{2\pi
i\varphi}\cdot\left(\begin{array}[]{cc}e^{u/2}e^{i\theta}&0\\\
0&e^{-u/2}e^{-i\theta}\end{array}\right)=\left(\begin{array}[]{cc}e^{u/2}e^{i\theta}e^{2\pi
i\varphi}&0\\\ 0&e^{-u/2}e^{-i\theta}e^{2\pi i\varphi}\end{array}\right).$

Following this procedure for every conjugacy class $C$ of length at most $R$
computes the spinc length spectrum for the lift $\widetilde{\Gamma}\subset G$
of $\Gamma$ corresponding to the pair
$s_{0}:\Gamma\rightarrow\mathrm{SL}_{2}(\mathbb{C})$ and
$\phi:\Gamma\rightarrow\mathbb{Q}/\mathbb{Z}.$

##  3\. 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.

###  3.1. Notation and conventions

  * •

Throughout the paper, we will use the convention for Fourier tranforms

$\widehat{H}(t)=\int_{\mathbb{R}}H(x)e^{-itx}dx.$

  * •

For a closed geodesic $\gamma,$ we denote by $\gamma_{0}$ a prime geodesic of
which $\gamma$ 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 $\ast d$ on $Y$ and
$\bar{Y}$ are opposite to each other. We use the identification
$\mathbb{H}^{3}=\mathrm{PSL_{2}}/\mathrm{PSU_{2}}$ (obtained by thinking of
the upper half plane model), for which the tangent space at $(0,1)$ is
$i\mathfrak{su}(2)$. We declare the (physicists’) Pauli matrices

$\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\quad\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right)\quad\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)$

to be a positively oriented basis.

###  3.2. Trace formula for functions

We begin with the classical trace formula for functions. Here, to each
eigenvalue

$0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\dots$

we associate the parameter $r_{n}\in\mathbb{R}\cup i[0,1]$ for which
$\lambda_{n}=1+r_{n}^{2}$ (in particular, $\lambda_{0}=i$). Non-zero
eigenvalues less than $1$ (i.e. such that $r_{n}\in i(0,1)$ ) are referred to
as small; the value $1$ is important because it is the bottom of the
$L^{2}$-spectrum of the Laplacian on $\mathbb{H}^{3}$ (see [14]).

######  Theorem 3.1.

For $H\in C^{\infty}_{c}(\mathbb{R})$ an even test function, the identity

$\sum_{n=0}^{\infty}\widehat{H}(r_{n})=\frac{\mathrm{vol}(Y)}{2\pi}\cdot\left(-H^{\prime\prime}(0)\right)+\sum_{[\gamma]\neq
1}\frac{\ell(\gamma_{0})}{|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|}H(\ell(\gamma))$

holds.

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

$0<\lambda_{1}^{*}\leq\lambda_{2}^{*}\leq\dots$

is of the form $\lambda_{n}^{*}=t_{j}^{2}$ for some $j$, where

$\dots\leq t_{-1}<0<t_{0}\leq t_{1}\leq t_{2}\leq\dots$

are the eigenvalues of $\ast d.$ 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 $H\in C^{\infty}_{c}(\mathbb{R})$ an even test function, the identity

$\frac{1}{2}(b_{1}(Y)-1)\cdot\widehat{H}(0)+\frac{1}{2}\sum\widehat{H}(t_{j})=\frac{\mathrm{vol(Y)}}{2\pi}\cdot(H(0)-H^{\prime\prime}(0))+\sum\ell(\gamma_{0})\frac{\cos(\mathrm{hol}(\gamma))}{|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|}H(\ell(\gamma)).$

holds. For $K\in C^{\infty}_{c}(\mathbb{R})$ an odd test function, the
identity

$\frac{1}{2}\sum\widehat{K}(t_{j})=i\sum\ell(\gamma_{0})\frac{\sin(\mathrm{hol}(\gamma))}{|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|}K(\ell(\gamma)).$

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 $\mathfrak{s}_{0}$ and consider
its twist by a character $\varphi$. The corresponding Dirac operator
$D_{B_{0}}$ has discrete spectrum

$\dots\leq s_{-1}<0\leq s_{0}\leq s_{1}\leq s_{2}\leq\dots$

unbounded in both directions. Also, recall that we introduced the notation
$\tilde{\gamma}$ for the lift of $\gamma$ corresponding to the base spin
structure $\mathfrak{s}_{0}$.

######  Theorem 3.3.

For $H\in C^{\infty}_{c}(\mathbb{R})$ an even test function, the identity

$\frac{1}{2}\sum\widehat{H}(s_{j})=\frac{\mathrm{vol(Y)}}{2\pi}\cdot\left(\frac{1}{4}H(0)-H^{\prime\prime}(0)\right)+\sum\ell(\gamma_{0})\frac{\cos(\mathrm{hol}(\tilde{\gamma}))\cos(\varphi({\gamma}))}{|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|}H(\ell(\gamma)).$

holds. For $K\in C^{\infty}_{c}(\mathbb{R})$ an odd test function, the
identity

$\frac{1}{2}\sum\widehat{K}(s_{j})=i\sum\ell(\gamma_{0})\frac{\sin(\mathrm{hol}(\tilde{\gamma}))\cos(\varphi({\gamma}))}{|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|}K(\ell(\gamma)).$

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
$\left(\mathbf{1}_{[-a,a]}\right)^{\ast 4}$, the fourth convolution power of
the indicator function of the interval $[-a,a]$. Rather than providing general
statements, let us point out two specific instances that will be used in this
work:

  1. (1)

the even trace formulas hold for $H(x)=\left(\mathbf{1}_{[-a,a]}\right)^{\ast
k}$, with $k\geq 6$ and the odd trace formulas hold for $K=H^{\prime}$. This
follows directly from the results in [28, Appendix $C$].

  2. (2)

the even trace formulas hold for a Gaussian $H(x)=e^{-x^{2}/2c}$, and the odd
trace formulas hold for $K=H^{\prime}$. 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.

##  4\. 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.

###  4.1. 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 $\eta$ invariant. Our account
loosely follows [14, Section $3.4$], suitably adapted to the simpler situation
we are dealing with. Given a continuous function
$f:\mathbb{R}^{>0}\rightarrow\mathbb{R}$, we define its Mellin transform to be

$M_{f}(s)=\int_{0}^{\infty}f(t)t^{s}\;\frac{dt}{t}.$

This is essentially the Fourier transform for the multiplicative group
$\mathbb{R}^{>0}$ equipped with the Haar measure $dt/t$. Assuming that
$f(t)=O(t^{-a})$ for $t\rightarrow 0$, and $f(t)=O(t^{-n})$ for
$t\rightarrow\infty$ for all $n$, we have that the integral absolutely
converges on the half-plane $\mathrm{Re}(s)>a$, 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

(6) $f(t)\sim\sum_{k=0}^{\infty}c_{k}t^{i_{k}}\quad\text{ for }t\rightarrow 0$

where $\\{i_{k}\\}$ is a strictly increasing sequence of real numbers with
$\lim i_{k}=+\infty$. Then $M_{f}$ extends to a meromorphic function on
$\mathbb{C}$ with a simple pole at $-i_{k}$ with residue $c_{k}$ for all $k$,
and holomorphic everywhere else. To see this, choose $L>0$ and let us write
$M_{f}(s)=M_{-}(s)+M_{+}(s)$ where

$\displaystyle M_{-}(s)$ $\displaystyle=\int_{0}^{L}f(t)t^{s}\;\frac{dt}{t},$
$\displaystyle M_{+}(s)$
$\displaystyle=\int_{L}^{\infty}f(t)t^{s}\;\frac{dt}{t}.$

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

$f(t)=\sum_{k=0}^{N-1}c_{k}t^{i_{k}}+r_{N}(t)$

with $r_{N}(t)=O(t^{i_{N}})$ for $t\rightarrow 0$. Integrating, for
$\mathrm{Re}(s)>-i_{N}$ we can therefore write

$\displaystyle M_{-}(s)$ $\displaystyle=\int_{0}^{L}f(t)t^{s}\;\frac{dt}{t}$
(7) $\displaystyle=\sum_{k=0}^{N-1}\frac{c_{k}\cdot
L^{s+i_{k}}}{s+i_{k}}+\int_{0}^{L}r_{N}(t)t^{s}\;\frac{dt}{t}.$

This expression provides a meromorphic continuation of $M_{-}$ to the half-
plane $\mathrm{Re}(s)>-i_{N}.$ This continuation has simple poles at
$-i_{0},\dots-i_{N-1}$ with residues $c_{0},\dots c_{N-1}$ and is holomorphic
everywhere else in the half-plane. Because our choice of $N$ was arbitrary,
this proves the claim.

######  Remark 4.1.

Though $M_{-}$ and $M_{+}$ both depend on $L,$ one readily checks that their
sum does not.

The above discussion readily generalizes to the case in which $f(t)=O(t^{-n})$
for some fixed $n$ when $t\rightarrow\infty$. In this case, the analytic
continuation (obtained again by breaking the integral
$M_{f}(s)=M_{-}(s)+M_{+}(s)$) defines a meromorphic continuation on the half-
plane $\mathrm{Re}(s)<n$.

###  4.2. The heat kernel

We briefly recall the classical analytic continuation of the eta function of
the odd signature operator

$\eta_{\mathrm{sign}}(s)=\sum_{t_{n}\neq
0}\frac{\mathrm{sgn}(t_{n})}{|t_{n}|^{s}}.$

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

(8)
$|a|^{-s}=\frac{1}{\Gamma(s)}\int_{0}^{\infty}e^{-|a|t}t^{s}\;\frac{dt}{t},$

where $\Gamma(s)=\int_{0}^{\infty}e^{-t}t^{s}\;\frac{dt}{t}$ is the classical
Gamma function.

In the simpler case of the Riemann zeta function, we have for
$\mathrm{Re}(s)>1$ (which ensures absolute convergence) the identity

$\displaystyle\zeta(s)$ $\displaystyle=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{1}{\Gamma(s)}\int_{0}^{\infty}e^{-nt}t^{s}\;\frac{dt}{t}$
$\displaystyle=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{1}{e^{t}-1}t^{s}\;\frac{dt}{t}.$

Recall that $\Gamma(s)$ has poles at $s=0,-1,-2,\dots$ with residue
$(-1)^{k}/k$ at $s=-k$. Given the asymptotic expansion for $t\rightarrow 0$ in
terms of Bernoulli numbers

$\frac{1}{e^{t}-1}\sim\sum_{k=-1}^{\infty}\frac{B_{k+1}}{(k+1)!}t^{k},$

and ${1}/({e^{t}-1})=O(t^{-n})$ for all $n$ when $t\rightarrow\infty$, we
recover the classical fact that $\zeta(s)$ 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
quantity222To be precise, they study the odd signature operator on coclosed
$1$-forms, which leads to an additional term involving $b_{1}(Y)$ in their
discussion.

$K(t)=-\sum\frac{\mathrm{sgn}(t_{n})}{2}\mathrm{erfc}(|t_{n}|\sqrt{t})$

where

$\mathrm{erfc}(t)=\frac{2}{\sqrt{\pi}}\int_{t}^{\infty}e^{-\xi^{2}}d\xi$

is the complementary error function. As $0$ is not an eigenvalue by
assumption, Weyl’s law (cf. Section 5) implies that $K(t)\rightarrow 0$
exponentially fast for $t\rightarrow\infty$. The derivative of $K(t)$ is
computed to be

$\displaystyle K^{\prime}(t)$ $\displaystyle=\frac{1}{\sqrt{4\pi t}}\sum
t_{n}e^{-t_{n}^{2}t}.$

Using $(\ref{gamma})$ and integrating by parts we get the formula

$\displaystyle\eta_{\mathrm{sign}}(2s)$
$\displaystyle=\frac{2\sqrt{\pi}}{\Gamma(s+1/2)}\int_{0}^{\infty}K^{\prime}(t)t^{s+1}\;\frac{dt}{t}$
(9)
$\displaystyle=-\frac{2s\sqrt{\pi}}{\Gamma(s+1/2)}\int_{0}^{\infty}K(t)t^{s}\;\frac{dt}{t}.$

The key point is that $K(t)$ admits an asymptotic expansion of the form

(10) $K(t)\sim\sum_{k\geq-n}a_{k}t^{k/2}\quad\text{ for }t\rightarrow 0,$

and therefore its Mellin transform has at worst a simple pole at $0$. In
$(\ref{etaheat})$, this pole is canceled by the term $s$ in the numerator, and
we see therefore that $\eta_{\mathrm{sign}}(s)$ is holomorphic near $0$. We
conclude by mentioning that the asymptotic expansion $(\ref{expansionK})$
follows by interpreting $K$ as the trace of the heat kernel on
$\mathbb{R}^{\geq 0}\times Y$ 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 $\ast d$ 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 $H^{\prime}$. Recalling that
with our convention

$\widehat{H^{\prime}}(t)=it\widehat{H}(t),$

we obtain the identity

(11)
$\sum_{n}t_{n}\widehat{H}(t_{n})=2\sum_{1\neq[\gamma]}\ell(\gamma_{0})\cdot\frac{\sin(\mathrm{hol}(\gamma))}{|1-e^{\mathbb{C}\ell(\gamma)}|\cdot|1-e^{-\mathbb{C}\ell(\gamma)}|}\cdot
H^{\prime}(\ell(\gamma)).$

For a fixed even test function $G$, and given $T>0$, we apply this to the
function $H(x)=G(x/T)$, for which $\widehat{H}(t)=T\widehat{G}(Tt)$, and
obtain the family of identities

(12)
$\sum_{n}Tt_{n}\widehat{G}(Tt_{n})=2\sum_{1\neq[\gamma]}\ell(\gamma_{0})\cdot\frac{\sin(\mathrm{hol}(\gamma))}{|1-e^{\mathbb{C}\ell(\gamma)}|\cdot|1-e^{-\mathbb{C}\ell(\gamma)}|}\cdot\frac{1}{T}G^{\prime}\left(\frac{\ell(\gamma)}{T}\right).$

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 $a\neq 0$ we
have the identity

$\displaystyle\int_{0}^{\infty}aT\widehat{G}(aT)T^{s-1}dT$
$\displaystyle=a\cdot\int_{0}^{\infty}\widehat{G}(aT)T^{s}dT$
$\displaystyle=a\cdot\int_{0}^{\infty}\widehat{G}(|a|T)T^{s}dT$
$\displaystyle=\frac{\mathrm{sgn}(a)}{|a|^{s}}\cdot\int_{0}^{\infty}\widehat{G}(T)T^{s}dT,$

where we used that $\widehat{G}$ is even in the second equality and performed
a change of variables in the last equality above. Therefore, for
$\mathrm{Re}(s)$ large enough (so that the sums converge absolutely), we have

$\displaystyle\eta_{\mathrm{sign}}(s)$
$\displaystyle=\sum_{t_{n}}\frac{\mathrm{sgn}(t_{n})}{|t_{n}|^{s}}$
$\displaystyle=\frac{\int_{0}^{\infty}\sum\left(t_{n}T\widehat{G}(t_{n}T)\right)T^{s}\;\frac{dT}{T}}{\int_{0}^{\infty}\widehat{G}(T)T^{s}dT}$
(13)
$\displaystyle=\frac{\int_{0}^{\infty}G_{T}T^{s}\;\frac{dT}{T}}{\int_{0}^{\infty}\widehat{G}(T)T^{s}dT}.$

We can recognize at the numerator and denominator the Mellin transforms of
$G_{T}$ and $T\widehat{G}(T)$ respectively. We have the following.

######  Lemma 4.1.

Suppose that $G$ is either:

  * •

a Gaussian function $x\mapsto e^{-x^{2}/2c}$.

  * •

$G$ is the $k$-th convolution power of an indicator function
$\frac{1}{2a}1_{[-a,a]}$, for some even $n\geq 6$.

Then the identity

$\eta_{\mathrm{sign}}(s)=\frac{\int_{0}^{\infty}G_{T}T^{s}\frac{dT}{T}}{\int_{0}^{\infty}T\widehat{G}(T)T^{s}\frac{dT}{T}}$

holds in an open domain in $\mathbb{C}$ 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 $T\rightarrow 0$ and $T\rightarrow\infty$. 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 $O(e^{2x}/2x)$ (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 $O(t^{3})$ (cf. Section 5), and $\widehat{G}$ is
still a Gaussian. By the properties of the Mellin transform discussed in
subsection 4.1, this implies that the numerator
$\int_{0}^{\infty}G_{T}T^{s}\frac{dT}{T}$ is an entire function of $s$. On the
other hand, the integrand in the denominator $T\widehat{G}(T)$ is rapidly
decaying for $T\rightarrow\infty$, 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 $\int_{0}^{\infty}\widehat{G}(T)dT>0$. The
claim then follows by uniqueness of the analytic continuation.

The case of $(\frac{1}{2a}1_{[-a,a]})^{\ast k}$, with $k\geq 6$ 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

$\widehat{G}(t)=\left(\frac{\sin(t)}{t}\right)^{k}.$

Looking at the spectral side, we see then that

$\lvert
G_{T}\lvert\leq\frac{1}{T^{k-1}}\cdot\sum_{n}\frac{1}{|t_{n}|^{k-1}}=O(T^{-k+1})$

for $T\rightarrow+\infty$. Furthermore $G_{T}$ is still rapidly decaying for
$T\rightarrow 0$, because $G$ is compactly supported on the geometric side.
The numerator is therefore a holomorphic function on the half-plane
$\mathrm{Re}(s)<k-1$. As in the case of Gaussians, $T\widehat{G}(T)$ has a
Taylor expansion at $0$ with only odd exponent terms, and
$\int_{0}^{\infty}T\widehat{G}(T)\frac{dT}{T}>0$; it is also $O(T^{-k+1})$ for
$T\rightarrow\infty$. The denominator is therefore a meromorphic function on
$\mathrm{Re}(s)<k-1$, 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 $L>0$, we have the formula

(14)
$\eta_{\mathrm{sign}}=\frac{\int_{0}^{L}G_{T}\;\frac{dT}{T}+\int_{L}^{\infty}G_{T}\;\frac{dT}{T}}{\int_{0}^{\infty}T\widehat{G}(T)\frac{dT}{T}}$

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:

$\displaystyle\int_{0}^{L}G_{T}\frac{dT}{T}$
$\displaystyle=\sum\ell(\gamma_{0})\cdot\frac{2\sin(\mathrm{hol}(\gamma))}{|1-e^{\mathbb{C}l(\gamma)}|\cdot|1-e^{-\mathbb{C}l(\gamma)}|}\int_{0}^{L}\frac{1}{T}G^{\prime}\left(\frac{\ell}{T}\right)\frac{dT}{T}$
$\displaystyle=\sum\ell(\gamma_{0})\cdot\frac{2\sin(\mathrm{hol}(\gamma))}{|1-e^{\mathbb{C}l(\gamma)}|\cdot|1-e^{-\mathbb{C}l(\gamma)}|}\cdot\frac{1}{\ell(\gamma)}\cdot\int_{\ell/L}^{\infty}yG^{\prime}(y)\frac{dy}{y}$
$\displaystyle=\sum\ell(\gamma_{0})\cdot\frac{2\sin(\mathrm{hol}(\gamma))}{|1-e^{\mathbb{C}l(\gamma)}|\cdot|1-e^{-\mathbb{C}l(\gamma)}|}\cdot\frac{1}{\ell(\gamma)}\cdot\int_{\ell/L}^{\infty}G^{\prime}(y)dy$
$\displaystyle=\sum\ell(\gamma_{0})\cdot\frac{2\sin(\mathrm{hol}(\gamma))}{|1-e^{\mathbb{C}l(\gamma)}|\cdot|1-e^{-\mathbb{C}l(\gamma)}|}\cdot\frac{1}{\ell(\gamma)}\cdot\left(-G\left(\frac{\ell(\gamma)}{L}\right)\right).$

and

$\displaystyle\int_{L}^{\infty}G_{T}\frac{dT}{T}$
$\displaystyle=\sum\int_{L}^{\infty}Tt_{n}\widehat{G}(Tt_{n})\frac{dT}{T}$
$\displaystyle=\sum\mathrm{sgn}(t_{n})\int_{L|t_{n}|}^{\infty}T\widehat{G}(T)\frac{dT}{T}$
$\displaystyle=\sum\mathrm{sgn}(t_{n})\int_{L|t_{n}|}^{\infty}\widehat{G}(T){dT},$

where we made a simple substitution in the integral.

Finally, the case of spinors follows in the same way by using the identity

$\displaystyle G_{T}$ $\displaystyle=\sum_{n}Ts_{n}\widehat{G}(Ts_{n})$
$\displaystyle=2\sum_{1\neq[\gamma]}\ell(\gamma_{0})\cdot\frac{\sin(\mathrm{hol}(\tilde{\gamma}))\cdot{\cos(\varphi_{\tilde{\gamma}})}}{|1-e^{\mathbb{C}\ell(\gamma)}|\cdot|1-e^{-\mathbb{C}\ell(\gamma)}|}\cdot\frac{1}{T}G^{\prime}\left(\frac{\ell(\gamma)}{T}\right),$

instead. The only additional observation is that the computation in (4.3)
still holds even when the Dirac operator has kernel. In fact, we have

$\displaystyle\eta_{\mathrm{Dir}}(s)$ $\displaystyle=\sum_{s_{n}\neq
0}\frac{\mathrm{sgn}(s_{n})}{|s_{n}|^{s}}$
$\displaystyle=\frac{\int_{0}^{\infty}\sum\left(s_{n}T\widehat{G}(s_{n}T)\right)T^{s}\;\frac{dT}{T}}{\int_{0}^{\infty}\widehat{G}(T)T^{s}dT}$
$\displaystyle=\frac{\int_{0}^{\infty}G_{T}T^{s}\;\frac{dT}{T}}{\int_{0}^{\infty}\widehat{G}(T)T^{s}dT}$

as is the second row the parameters $s_{n}$ equal to zero do not contribute.

##  5\. 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 $(X,g)$, the asymptotic

(15) $\\#\\{\text{eigenvalues $\lambda$ of $\Delta_{g}$ with
$\sqrt{\lambda}\leq T$}\\}\sim C_{n}\cdot\mathrm{vol}_{g}(X)\cdot T^{n}.$

holds, where $\Delta_{g}$ is the Hodge Laplacian acting on functions.
Analogous versions hold more generally for squares of Dirac type operators,
such as the Hodge Laplacian $(d+d^{*})^{2}$ acting on $k$-forms or the Dirac
Laplacian $D_{B_{0}}^{2}$, see [4]. From Equation (15), one expects the
following local version of Weyl law to hold

$\\#\\{\text{eigenvalues $\lambda$ of $\Delta_{g}$ with
$\sqrt{\lambda}\in[T,T+1]$}\\}\sim n\cdot C_{n}\cdot\mathrm{vol}_{g}(X)\cdot
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

(16) $\\#\\{\text{eigenvalues $\lambda$ of $\Delta_{g}$ with
$\sqrt{\lambda}\in[T,T+1]$}\\}\leq A\cdot T^{n-1}+B$

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
$C^{\prime}_{n}\cdot\mathrm{vol}_{g}(X).$

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 $\mathrm{inj}>0.15$ and $\mathrm{vol}<6.5$, 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 $R>0$ (which will later be a given lower bound on the
injectivity radius). Choose an even function $\varphi$ with support in
$[-2,2]$ and non-negative Fourier transform. Again, our convention is

$\widehat{\varphi}(t)=\int\varphi(x)e^{-ixt}dt.$

To simplify our discussion, we will make the following:

Assumption. We will assume for simplicity throughout the section that both
$\varphi$ and $\widehat{\varphi}$ achieve their maximum at $0$, and that the
minimum $m_{R}$ of $\widehat{\varphi}$ in $[-R,R]$ is achieved at $\pm R$.

######  Example 5.1.

For the specialization $R=0.15,$ we will choose $\varphi_{0}=\beta\ast\beta$,
where

$\beta(x)=\begin{cases}e^{-1/(1-x^{2})}\text{ if }|x|<1\\\ 0\text{
otherwise.}\end{cases}$

This function satisfies the assumptions, and $m_{R}\sim 0.19643$.

######  Remark 5.1.

Of course, this implies that $\varphi_{0}$ satisfies the assumption also for
$R<0.15$. To obtain a function that satisfies the assumption for large values
of $R$, we can look at functions of the form $\varphi_{0}(Cx)$.

Consider $\varphi_{R}(x)=\varphi\left(\frac{x}{R}\right)$, which is supported
in $[-2R,2R]$. We have

$\displaystyle\varphi_{R}(0)$ $\displaystyle=\varphi(0),$
$\displaystyle\varphi_{R}^{\prime\prime}(0)$
$\displaystyle=\frac{1}{R^{2}}\varphi^{\prime\prime}(0),$

and

$\displaystyle\widehat{\varphi_{R}}(t)$
$\displaystyle=\int\varphi\left(\frac{x}{R}\right)e^{-ixt}dx$
$\displaystyle=\int\varphi(x^{\prime})e^{iRx^{\prime}t}Rdx^{\prime}$
$\displaystyle=R\widehat{\varphi}(Rt).$

Consider also

$\displaystyle\varphi_{R,\nu}$ $\displaystyle=\varphi_{R}(x)(e^{i\nu
x}+e^{-i\nu x})$ $\displaystyle=2\varphi_{R}(x)\cos(\nu x).$

Then

$\displaystyle\varphi_{R,\nu}(0)$ $\displaystyle=2\varphi(0),$
$\displaystyle\varphi_{R,\nu}^{\prime\prime}(0)$
$\displaystyle=2\left(\frac{1}{R^{2}}\varphi^{\prime\prime}(0)-\nu^{2}\varphi(0)\right),$

and

$\widehat{\varphi_{R,\nu}}(t)=\widehat{\varphi_{R}}(t+\nu)+\widehat{\varphi_{R}}(t-\nu).$

###  5.2. Local Weyl law for coexact $1$-forms

We begin with the case of coexact $1$-forms, which is the simplest to analyze.
For $\nu\geq 0$, denote by $\delta^{*}(\nu)$ the number of spectral parameters
$t_{j}$ with $|t_{j}|\in[\nu,\nu+1]$. We evaluate the trace formula in Theorem
3.2 for the test function $\varphi_{R,\nu}(x)$ with $R$ less than the
injectivity radius of $Y$. As $\varphi_{R,\nu}$ is supported in $[-2R,2R]$,
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

$\frac{\mathrm{vol}(Y)}{2\pi}\left(\varphi_{R,\nu}(0)-\varphi_{R,\nu}^{{}^{\prime\prime}}(0)\right)+\frac{1}{2}\widehat{\varphi_{R,\nu}}(0)=\sum_{j}\frac{1}{2}\widehat{\varphi_{R,\nu}}(|t_{j}|).$

By the identities of the previous subsection we therefore obtain

$\displaystyle\frac{\mathrm{vol}(Y)}{\pi}\left(\varphi(0)-\frac{1}{R^{2}}\varphi^{\prime\prime}(0)+\nu^{2}\varphi(0)\right)+{R}\widehat{\varphi}(R\nu)$
$\displaystyle=\frac{1}{2}\sum_{j}\widehat{\varphi_{R}}(|t_{j}|+\nu)+\widehat{\varphi_{R}}(|t_{j}|-\nu)$
$\displaystyle\geq\sum_{j}\frac{1}{2}\widehat{\varphi_{R}}(|t_{j}|-\nu)$
$\displaystyle\geq\frac{1}{2}R\cdot m_{R}\cdot\delta^{*}(\nu)$

using first that $\widehat{\varphi}_{R}$ is non-negative, and then that

$\displaystyle\widehat{\varphi_{R}}(t-\nu)$
$\displaystyle=R\cdot\widehat{\varphi}(R\cdot(t-\nu))$ $\displaystyle\geq
R\cdot m_{R}$

for $t\in[\nu,\nu+1]$. For any choice of suitable test function $\varphi$,
this provides the following upper bound on $\delta^{*}(\nu)$ in terms of
$\mathrm{vol}(Y)$ and injectivity radius:

$\delta^{*}(\nu)\leq\left(\frac{2\cdot\mathrm{vol(Y)}\cdot\varphi(0)}{\pi
R\cdot m_{R}}\right)\nu^{2}+\frac{2}{R\cdot
m_{R}}\left(R\widehat{\varphi}(0)+\frac{\mathrm{vol}(Y)}{\pi}\left(\varphi(0)-\frac{1}{R^{2}}\varphi^{\prime\prime}(0)\right)\right)$

where we used that $\widehat{\varphi}$ achieve its maximum at $0$. Working out
the computations using our test function $\varphi_{0}$, we obtain the
following.

######  Proposition 5.1.

Let $Y$ be a hyperbolic rational homology sphere with $\mathrm{vol}(Y)<6.5$
and $\mathrm{inj}(Y)>0.15$. Then the inequality

$\delta^{*}(\nu)\leq 18.7\nu^{2}+2577.3$

holds.

######  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
$\mu>0$ for $Y$. Recall that for such a number, the set of points of $Y$ with
local injectivity radius $<\mu/2$ is the disjoint union of tubes around the
finitely many closed geodesics with length $<\mu$. For example, in [33] it is
shown $\mu=0.1$ is a Margulis number for all closed oriented hyperbolic three-
manifolds. As the number of closed geodesics shorter than $\mu$ can be bounded
above in terms of the volume [18], one obtains estimates for the spectral
density by considering a test function supported in $[-\mu,\mu]$. We will not
pursue the exact output of this approach in the present work.

###  5.3. 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 $\delta(\nu)$ to be the number of parameters in $[\nu,\nu+1]$,
and let $\delta_{s}$ 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

$\sum\widehat{\varphi_{R,\nu}}(r_{n})=-\frac{\mathrm{vol}(Y)}{2\pi}\varphi_{R,\nu}^{\prime\prime}(0)$

Here we recall that $r_{n}\in\mathbb{R}^{\geq 0}\cup i[0,1]$ corresponds to
the eigenvalue $\lambda_{n}=1+r_{n}^{2}$.

We start by bounding the number of small eigenvalues. For this purpose, we set
$\nu=0$. For real $t$, we have

$\displaystyle\widehat{\varphi}(it)$
$\displaystyle=\int_{\mathbb{R}}\varphi(x)e^{-tx}dx$
$\displaystyle=\int_{\mathbb{R}}\varphi(x)\cosh(tx)dx$

which as a function of $t\in\mathbb{R}$ is non-negative, even, convex, and has
minimum at $0$. Therefore we have

$\displaystyle-\frac{\mathrm{vol}(Y)\varphi^{\prime\prime}(0)}{2\pi R^{2}}$
$\displaystyle=\sum\widehat{\varphi_{R}}(r_{n})$
$\displaystyle\geq\sum_{r_{n}\text{ imaginary}}\widehat{\varphi_{R}}(r_{n})$
$\displaystyle=R\cdot\sum_{r_{n}\text{
imaginary}}\widehat{\varphi}(Rr_{n})\hskip 14.22636pt$ $\displaystyle\geq
R\widehat{\varphi}(0)\cdot\delta_{s},$

so that

$\delta_{s}\leq-\frac{\mathrm{vol}(Y)\varphi^{\prime\prime}(0)}{2\pi
R^{3}\widehat{\varphi}(0)}.$

To bound large eigenvalues, we look at $\widehat{\varphi_{R,\nu}}$ for
$\nu\neq 0$. Unfortunately, this is not necessarily positive on the imaginary
axis. On the other hand, we have for $t\in[0,1]$

$\displaystyle|\widehat{\varphi_{R,\nu}}(it)|$
$\displaystyle=\left|2\int_{-2R}^{2R}\varphi(x/R)\cos(\nu
x)e^{-xt}dx\right|\hskip 14.22636pt(\text{because }\varphi\text{ is supported
on }[-2,2])$ $\displaystyle=2R\left|\int_{-2}^{2}\varphi(y)\cos(\nu
Ry)e^{-Ryt}dy\right|$ $\displaystyle\leq 2R\varphi(0)\int_{-2}^{2}e^{-Rty}dy$
$\displaystyle=2\varphi(0)\cdot\frac{e^{2Rt}-e^{-2Rt}}{t}$ $\displaystyle\leq
2\varphi(0)(e^{2R}-e^{-2R})$

where the last inequality holds because $(e^{2Rt}-e^{-2Rt})/t$ is increasing
for $t\in[0,1]$. The trace formula then implies

$\displaystyle\frac{\mathrm{vol(Y)}}{\pi}\left(\nu^{2}\varphi(0)-\frac{1}{R^{2}}\varphi^{\prime\prime}(0)\right)+2\varphi(0)\cdot(e^{2R}-e^{-2R})\delta_{s}$
$\displaystyle\geq\sum_{\text{real}}\widehat{\varphi_{R,\nu}}(r_{n})$
$\displaystyle\geq\sum_{\text{real}}\widehat{\varphi_{R}}(r_{n}-\nu)$
$\displaystyle\geq R\cdot m_{R}\cdot\delta(\nu)$

so that

$\delta(\nu)\leq\left(\frac{\mathrm{vol(Y)}\cdot\varphi(0)}{\pi R\cdot
m_{R}}\right)\nu^{2}+\frac{1}{R\cdot
m_{R}}\left(-\frac{\mathrm{vol(Y)}\varphi^{\prime\prime}(0)}{\pi
R^{2}}+2\varphi(0)\cdot({e^{2R}-e^{-2R}})\delta_{s}\right).$

Concretely, using again the test function $\varphi_{0}$, we obtain.

######  Proposition 5.2.

Let $Y$ be a hyperbolic rational homology sphere with $\mathrm{vol}(Y)<6.5$
and $\mathrm{inj}(Y)>0.15$. Then the inequalities

$\displaystyle\delta_{s}$ $\displaystyle\leq 637$ $\displaystyle\delta(\nu)$
$\displaystyle\leq 9.4\nu^{2}+4782$

hold.

###  5.4. Local Weyl law for spinors

This case is essentially the same as that of coexact $1$-forms. For a given
spinc structure, denote by $\delta^{D}(\nu)$ the number of Dirac eigenvalues
with absolute value in $[\nu,\nu+1]$. Choosing again $R=\mathrm{inj}(Y)$, we
have

$\frac{\mathrm{vol}(Y)}{2\pi}\left(\frac{1}{4}\varphi_{R,\nu}(0)-\varphi_{R,\nu}^{{}^{\prime\prime}}(0)\right)=\sum_{j}\frac{1}{2}\widehat{\varphi_{R,\nu}}(s_{j}).$

so that

$\frac{\mathrm{vol}(Y)}{\pi}\left(\frac{1}{4}\varphi(0)-\frac{1}{R^{2}}\varphi^{\prime\prime}(0)+\nu^{2}\varphi(0)\right)\geq\frac{1}{2}R\cdot
m_{R}\cdot\delta^{D}(\nu)$

and

$\delta^{D}(\nu)\leq\left(\frac{2\mathrm{vol}(Y)\cdot\varphi(0)}{\pi R\cdot
m_{R}}\right)\nu^{2}+\frac{2\mathrm{vol(Y)}}{\pi R\cdot
m_{R}}\left(\frac{1}{4}\varphi(0)-\frac{1}{R^{2}}\varphi^{\prime\prime}(0)\right).$

Specializing using the test function $\varphi_{0},$ we obtain:

######  Proposition 5.3.

Let $Y$ be a hyperbolic rational homology sphere with $\mathrm{vol}(Y)<6.5$
and $\mathrm{inj}(Y)>0.15$. Then for any spinc structure, the inequality

$\delta^{D}(\nu)\leq 18.7\nu^{2}+2561.3$

holds.

##  6\. 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

$\displaystyle\delta^{*}(\nu)$ $\displaystyle\leq A\nu^{2}+B\hskip
17.07182pt(\text{coexact 1-forms})$ $\displaystyle\delta_{s}$
$\displaystyle\leq C\hskip 49.79231pt(\text{0-form eigenvalues}<1)$
$\displaystyle\delta(\nu)$ $\displaystyle\leq\frac{A}{2}\nu^{2}+D\hskip
14.22636pt(\text{0-form eigenvalues}\geq 1)$ $\displaystyle\delta^{D}(\nu)$
$\displaystyle\leq A\nu^{2}+E.\hskip 14.22636pt(\text{spinor eigenvalues})$

where the specific values of the constants for a manifold with
$\mathrm{vol}<6.5$ and $\mathrm{inj}>0.15$ were determined in the previous
section.

###  6.1. Bounds on $\eta_{\mathrm{sign}}$.

Recall from Subsection 4 (after setting $L=1$): for an arbitrary admissible
test function $G$,

$\eta_{\mathrm{sign}}=\frac{\int_{0}^{1}G_{T}\frac{dT}{T}+\int_{1}^{\infty}G_{T}\frac{dT}{T}}{\int_{0}^{\infty}\widehat{G}(T)dT}$

where

$\displaystyle G_{T}$ $\displaystyle=\sum_{n}Tt_{n}\widehat{G}(Tt_{n})$
$\displaystyle=2\sum_{1\neq[\gamma]}\ell(\gamma_{0})\cdot\frac{\sin(\mathrm{hol}(\gamma))}{|1-e^{\mathbb{C}\ell(\gamma)}|\cdot|1-e^{-\mathbb{C}\ell(\gamma)}|}\cdot\frac{1}{T}G^{\prime}\left(\frac{\ell(\gamma)}{T}\right).$

For our purposes, it is convenient to restrict our attention to
$G(x)=(\frac{1}{2}\mathbf{1}_{[-1,1]})^{\ast k}$ for even $k\geq 6$. This
function has support in $[-k,k]$, and its Fourier transform is
$\widehat{G}(t)=\mathrm{sinc}^{k}(t)$ where $\mathrm{sinc}(t):=\sin(t)/t$. We
will denote

$\displaystyle c_{k}:=\int_{0}^{\infty}\mathrm{sinc}^{k}(t)\;dt$
$\displaystyle d_{k}:=\int_{1}^{\infty}\mathrm{sinc}^{k}(t)\;dt.$

Let us consider the numerator of the expression for $\eta_{\mathrm{sign}}$; 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

$\displaystyle\int_{1}^{\infty}G_{T}\frac{dT}{T}$
$\displaystyle=\int_{1}^{\infty}\sum_{t_{n}}Tt_{n}\cdot\mathrm{sinc}^{k}(Tt_{n})\frac{dT}{T}$
$\displaystyle=\int_{1}^{\infty}\sum_{t_{n}}\mathrm{sgn}(t_{n})\cdot
T|t_{n}|\cdot\mathrm{sinc}^{k}(T|t_{n}|)\frac{dT}{T}$
$\displaystyle=\sum_{t_{n}}\mathrm{sgn}(t_{n})\cdot\int_{|t_{n}|}^{\infty}\mathrm{sinc}^{k}(T)dT.$

We split the last sum in two parts $\sum_{|t_{n}|\leq 2}+\sum_{|t_{n}|>2}$. In
absolute value, the first part can be bounded above as

$\displaystyle\sum_{|t_{n}|\leq
2}\int_{|t_{n}|}^{\infty}\mathrm{sinc}^{k}(T)dT$
$\displaystyle\leq\delta^{*}(0)\cdot\int_{0}^{\infty}\mathrm{sinc}^{k}(T)dT+\delta^{*}(1)\cdot\int_{1}^{\infty}\mathrm{sinc}^{k}(T)dT$
$\displaystyle=c_{k}\cdot B+d_{k}\cdot(A+B).$

For the second part, using $\mathrm{sinc}^{k}(T)\leq T^{-k}$, we have

$\displaystyle\sum_{|t_{n}|\geq
2}\int_{|t_{n}|}^{\infty}\mathrm{sinc}^{k}(T)dT$
$\displaystyle\leq\sum_{|t_{n}|\geq
2}\int_{|t_{n}|}^{\infty}\frac{1}{T^{k}}dT$
$\displaystyle=\frac{1}{k-1}\sum_{|t_{n}|\geq 2}\frac{1}{|t_{n}|^{k-1}}$
$\displaystyle=\frac{1}{k-1}\sum_{m=2}^{\infty}\sum_{|t_{n}|\in[m,m+1]}\frac{1}{|t_{n}|^{k-1}}$
$\displaystyle\leq\frac{1}{k-1}\sum_{m=2}^{\infty}\delta^{*}(m)\frac{1}{m^{k-1}}$
$\displaystyle\leq\frac{1}{k-1}\sum_{m=2}^{\infty}(Am^{2}+B)\frac{1}{m^{k-1}}$
$\displaystyle=\frac{1}{k-1}\left(A(\zeta(k-3)-1)+B(\zeta(k-1)-1)\right)$

and therefore

$\left|\int_{1}^{\infty}G_{T}\frac{dT}{T}\right|\leq B\cdot c_{k}+(A+B)\cdot
d_{k}+\frac{1}{k-1}\left(A(\zeta(k-3)-1)+B(\zeta(k-1)-1)\right).$

For the geometric side, using a simple substitution, we have

$\displaystyle\int_{0}^{1}G_{T}\frac{dT}{T}$
$\displaystyle=2\int_{0}^{1}\sum\ell(\gamma_{0})\frac{\sin(\mathrm{hol}(\gamma))}{|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|}\frac{G^{\prime}\left(\frac{\ell(\gamma)}{T}\right)}{T}\frac{dT}{T}$
$\displaystyle=-2\sum\ell(\gamma_{0})\frac{\sin(\mathrm{hol}(\gamma))}{|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|}\frac{G(\ell(\gamma))}{\ell(\gamma)}$

As $\ell(\gamma)\geq 2\mathrm{inj}(Y)$, we have by taking absolute values

$\left|\int_{0}^{1}G_{T}\frac{dT}{T}\right|\leq\frac{1}{\mathrm{inj}(Y)}\sum\ell(\gamma_{0})\frac{1}{|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|}{G(\ell(\gamma))}.$

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

$\displaystyle\sum\ell(\gamma_{0})\frac{1}{|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|}{G(\ell(\gamma))}$
$\displaystyle=\frac{\mathrm{vol}(Y)}{2\pi}G^{\prime\prime}(0)+\sum_{r_{n}}\widehat{G}(r_{n})$
$\displaystyle\leq\sum_{r_{n}}\widehat{G}(r_{n})$

where we used that $G^{\prime\prime}(0)$ is negative. We deal with imaginary
and real parameters separately. For imaginary parameters, we have

$\widehat{G}(it)=\mathrm{sinc}^{k}(it)=\frac{\sinh^{k}(t)}{t^{k}}$

which is increasing in $t$ and therefore we obtain the inequality

$\displaystyle\sum_{\text{imaginary }r_{n}}\widehat{G}(r_{n})$
$\displaystyle\leq\sinh(1)^{k}\cdot\delta_{s}$ $\displaystyle\leq
C\cdot\sinh(1)^{k}.$

For the real parameters, using $\widehat{G}\leq 1$ and $\widehat{G}\leq
1/x^{k}$ respectively we have

$\displaystyle\sum_{\text{real }r_{n}}\widehat{G}(r_{n})$
$\displaystyle=\sum_{r_{n}\in[0,1]}\widehat{G}(r_{n})+\sum_{m=1}^{\infty}\sum_{r_{n}\in[m,m+1]}\widehat{G}(r_{n})$
$\displaystyle\leq\delta(0)\cdot
1+\sum_{m=1}^{\infty}\delta(m)\cdot\frac{1}{m^{k}}$ $\displaystyle\leq
D+\sum_{m=1}^{\infty}\frac{(A/2)m^{2}+D}{m^{k}}$
$\displaystyle=\frac{A}{2}\zeta(k-2)+D(\zeta(k)+1).$

Hence, putting everything in §6.1 together, we have

$\left|\int_{0}^{1}G_{T}\frac{dT}{T}\right|\leq\frac{1}{\mathrm{inj}(Y)}\left(C\cdot\sinh(1)^{k}+\frac{A}{2}\zeta(k-2)+D(\zeta(k)+1)\right).$

Putting everything together, we obtain the following.

######  Proposition 6.1.

For every even $k\geq 6$ the inequality

$\displaystyle\lvert\eta_{\mathrm{sign}}(Y)\lvert\leq\frac{1}{c_{k}}\bigg{(}B\cdot
c_{k}+(A+B)\cdot d_{k}+\frac{1}{k-1}(A(\zeta(k-3)-1)+B(\zeta(k-1)-1))$
$\displaystyle+\frac{1}{\mathrm{inj}(Y)}(C\cdot\sinh(1)^{k}+\frac{A}{2}\zeta(k-2)+D(\zeta(k)+1))\bigg{)}$

holds, 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 $\mathrm{vol}(Y)<6.5$ and
$\mathrm{inj}(Y)>0.15$, then $\lvert\eta_{\mathrm{sign}}(Y)\lvert\leq 108267$.

###  6.2. Bounds on $\eta_{\mathrm{Dir}}$

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

$\displaystyle G_{T}$ $\displaystyle=\sum_{n}Ts_{n}\widehat{G}(Ts_{n})$
$\displaystyle=2\sum_{1\neq[\gamma]}\ell(\gamma_{0})\cdot\frac{\sin(\mathrm{hol}(\tilde{\gamma}))\cdot{\cos(\varphi_{\tilde{\gamma}})}}{|1-e^{\mathbb{C}\ell(\gamma)}|\cdot|1-e^{-\mathbb{C}\ell(\gamma)}|}\cdot\frac{1}{T}G^{\prime}\left(\frac{\ell(\gamma)}{T}\right).$

Here we will again bound the integral $\int_{1}^{\infty}G_{T}\frac{dT}{T}$
using the bound for the spectral density $\delta^{D}(\nu)$, and the integral
$\int_{0}^{1}G_{T}\frac{dT}{T}$ using the trace formula for functions. The
final result is the following:

######  Proposition 6.3.

For every even $k\geq 6$ the inequality

$\displaystyle\lvert\eta_{\mathrm{Dir}}(Y)\lvert\leq\frac{1}{c_{k}}\bigg{(}E\cdot
c_{k}+(A+E)\cdot d_{k}+\frac{1}{k-1}(A(\zeta(k-3)-1)+E(\zeta(k-1)-1))$
$\displaystyle+\frac{1}{\mathrm{inj}(Y)}(C\cdot\sinh(1)^{k}+\frac{A}{2}\zeta(k-2)+D(\zeta(k)+1))\bigg{)}$

holds for all spinc structures.

Setting again $k=8$, we have the following effective estimate.

######  Corollary 6.4.

If $Y$ is a hyperbolic rational homology sphere with $\mathrm{vol}(Y)<6.5$ and
$\mathrm{inj}(Y)>0.15$, then for every spinc structure
$\lvert\eta_{\mathrm{Dir}}(Y)\lvert\leq 108249$.

###  6.3. Proof of Theorem 0.1

Given our discussion, Theorem 0.1 follows directly from the fact that for a
minimal hyperbolic $L$-space

$h(Y,\mathfrak{s})=-\frac{\eta_{\mathrm{sign}}}{8}-\frac{\eta_{\mathrm{Dir}}}{2},$

see Proposition 1.1.

##  7\. An explicit example: the Weeks manifold

In order to prove Theorem 0.1, we used test functions supported in
$[-2\cdot\mathrm{inj},2\cdot\mathrm{inj}]$, 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 $\mathsf{W}$; a similar approach works also for other small volume
hyperbolic minimal $L$-spaces in the Hodgson-Weeks census we discussed in
[28].

Recall that $H_{1}(\mathsf{W},\mathbb{Z})=(\mathbb{Z}/5\mathbb{Z})^{2}$. 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.

######  Proposition 7.1.

The unique spin structure on $\mathsf{W}$ has Frøyshov invariant $-1/2$. For
the remaining $24$ spinc structures the Frøyshov invariant either $\pm 1/10$,
or $\pm 3/10$, and each value is the invariant of exactly six spinc
structures.

This is essentially showed in [36], using the fact that $\mathsf{W}$ 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

$\eta_{\mathrm{sign}}=0.040028711\dots$

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 spinc
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 spinc
structure $\mathfrak{s}$ we determine an explicit function

$J_{\mathfrak{s}}:\mathbb{R}^{\geq 0}\rightarrow\mathbb{R}^{\geq 0}$

with the property that if $\pm s$ are eigenvalues of the Dirac operator whose
multiplicities add to $m$, then $J_{\mathfrak{s}}(|s|)\geq m$. The pictures in
the range $[0,6]$ can be found in Figure 1; Class $1$ is the class of the spin
structure, and the remaining $24$ spinc 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 spinc structures [32]. We will exploit symmetry under the isometry
group in the more complicated case of $\mathsf{SW}$ in Section 10.

(a) Class $1$ (the spin structure)

(b) Class $2$

(c) Class $3$

(d) Class $4$

(e) Class $5$

Figure 1. Pictures for the Weeks manifold

We compute an approximation to $\eta_{\mathrm{Dir}}$ 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 $\leq 6.5$, 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 works well because there is a good spectral gap (but
increases of course the error coming from truncating the sum over the
geodesics).

Using the formula (14) with $L=1$, we obtained the following approximate
values for $h=-\eta_{\mathrm{sign}}/8-\eta_{\mathrm{Dir}}/2$:

Spinc class | Approximate value of $h$
---|---
1 | $-0.49997\dots$
2 | $0.09971\dots$
3 | $-0.09994\dots$
4 | $0.29994\dots$
5 | $-0.29994\dots$

which are very close to the exact values in Proposition 7.1.

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. (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 $(2n+1)/10$. Given this, one only needs to
prove that the errors are less than $\frac{1}{20}$ 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. (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]).

  3. (3)

To bound the error on the spectral side, the key observation is that large
eigenvalues contribute very little to the error. For example (thinking about
Classes $1$, $4$ and $5$), picking $G(x)=e^{-x^{2}/2}$ (for which
$\widehat{G}(t)=\sqrt{2\pi}e^{-x^{2}/2}$ and
$\int_{0}^{\infty}\widehat{G}(t)dt=\pi$) and $L=1$, an eigenvalue with
$|t|>3.5$ contributes to the eta invariant at most

$\frac{1}{\pi}\int_{3.5}^{\infty}\sqrt{2\pi}e^{-x^{2}/2}dx=0.00046\dots$

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 $(2)$ and $(3)$, the error
estimate can be improved by computing a larger portion of the (spinc) length
spectrum.

######  Remark 7.2.

Let us point out that combining Propositions 7.1 and 1.1 we obtain a formula
for the Dirac eta invariants $\eta_{\mathrm{Dir}}$ of the Weeks manifold in
terms of the signature eta invariant $\eta_{\mathrm{sign}}$. In particular,
this allows us to provide a computation of $\eta_{\mathrm{Dir}}$ which is as
precise as the one provided by Snap; for example, as mentioned in the
introduction we have

$\eta_{\mathrm{Dir}}=0.989992\dots$

for the unique spin structure on the Weeks manifold.

##  8\. The linking form of the Seifert-Weber dodecahedral space

In this section we compute the linking form of the Seifert-Weber dodecahedral
space $\mathsf{SW}$. 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

$Q:H_{1}(Y,\mathbb{Z})\times
H_{1}(Y,\mathbb{Z})\rightarrow\mathbb{Q}/\mathbb{Z}$

defined geometrically as follows: given elements $x,y$, choose $n$ for which
$ny=0$ and pick a chain $T$ with $\partial T=ny$. Then $Q(x,y)$ is the
intersection number of $x$ and $T$, divided by $n$. More abstractly, the
corresponding map

$Q^{\hash}:H_{1}(Y,\mathbb{Z})\rightarrow\mathrm{Hom}(H_{1}(Y,\mathbb{Z}),\mathbb{Q}/\mathbb{Z})\cong
H^{1}(Y,\mathbb{Q}/\mathbb{Z})$

is the composition

$H_{1}(Y,\mathbb{Z})\cong H^{2}(Y,\mathbb{Z})\cong
H^{1}(Y,\mathbb{Q}/\mathbb{Z})$

where the first isomorphism is Poincaré duality and the second is the
Bockstein homomorphism for the short exact sequence of coefficients

$0\rightarrow\mathbb{Z}\rightarrow\mathbb{Q}\rightarrow\mathbb{Q}/\mathbb{Z}\rightarrow
0.$

In particular, $Q^{\hash}$ is an isomorphism and $Q$ is non-degenerate.

######  Proposition 8.1.

For the explicit basis $a,b,c$ of
$H_{1}(\mathsf{SW})\cong(\mathbb{Z}/5\mathbb{Z})^{3}$ identified in the proof
below, the linking form $Q_{\mathsf{SW}}$ is represented by the matrix

$\left(\begin{array}[]{ccc}0&2/5&3/5\\\ 2/5&0&2/5\\\
3/5&2/5&0\end{array}\right).$

######  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 $\mathsf{SW}$ we
will discuss in the upcoming sections works more generally.

###### Proof.

Recall that $\mathsf{SW}$ 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: $1\leftrightarrow 12$,
$2\leftrightarrow 9$, $3\leftrightarrow 10$, $4\leftrightarrow 11$,
$5\leftrightarrow 7$, $6\leftrightarrow 8$.

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:

$\displaystyle(\partial F_{1})$ $\displaystyle\hskip 25.6073pta+b+c+d+e$
$\displaystyle=0$ $\displaystyle(\partial F_{2})$ $\displaystyle\hskip
12.80365pt-a-b+c\hskip 19.91684pt+e+f$ $\displaystyle=0$
$\displaystyle(\partial F_{3})$ $\displaystyle\hskip 25.6073pta-b-c+d\hskip
19.91684pt+f$ $\displaystyle=0$ $\displaystyle(\partial F_{4})$
$\displaystyle\hskip 44.10185ptb-c-d+e+f$ $\displaystyle=0$
$\displaystyle(\partial F_{5})$ $\displaystyle\hskip 25.6073pta\hskip
17.07182pt+c-d-e+f$ $\displaystyle=0$ $\displaystyle(\partial F_{6})$
$\displaystyle\hskip 12.80365pt-a+b\hskip 17.07182pt+d-e+f$ $\displaystyle=0$

Simple elementary row operations reduce this system to

$\displaystyle d$ $\displaystyle=a+2b+3c$ $\displaystyle e$
$\displaystyle=3a+2b+c$ $\displaystyle f$ $\displaystyle=3a+4b+3c$

and

(17) $5a=5b=5c=0$

We therefore see that $a,b,c$ generate the first homology
$H_{1}(\mathsf{SW})\cong(\mathbb{Z}/5)^{3}$. Furthermore, we can interpret the
equations (17) as the geometric identities

$\displaystyle 5a$ $\displaystyle=\partial(F_{1}-F_{2}+F_{3}+F_{5}-F_{6})$
$\displaystyle 5b$ $\displaystyle=\partial(F_{1}-F_{2}-F_{3}+F_{4}+F_{6})$
$\displaystyle 5c$ $\displaystyle=\partial(F_{1}+F_{2}-F_{3}-F_{4}+F_{5}).$

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

$\displaystyle A$ $\displaystyle=4a+2b+4c$ $\displaystyle B$
$\displaystyle=3a+b+4c$ $\displaystyle C$ $\displaystyle=4a+b+3c$

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 $a,b,c$.

lk | a | b | c
---|---|---|---
A | $1/5$ | $1/5$ | $1/5$
B | $-1/5$ | $-1/5$ | $1/5$
C | $1/5$ | $-1/5$ | $-1/5$

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 $Q_{\mathsf{SW}}(x,x)$ for
$x\neq 0$ attains possible values as in the following table.

$Q_{\mathsf{SW}}(x,x)$ | number of $x\neq 0$ 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
$\mathsf{SW}$. 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

$\mathrm{Isom}(\mathsf{SW})\hookrightarrow\mathrm{O}(Q_{\mathsf{SW}})$

has image contained in $\mathrm{SO}(Q_{\mathsf{SW}})$. The latter has $120$
elements, and we get therefore an isomorphism
$\mathrm{Isom}(\mathsf{SW})\cong\mathrm{SO}(Q_{\mathsf{SW}})$. We have
therefore the following.

######  Corollary 8.2.

Consider two spinc structures $\mathfrak{s}$, $\mathfrak{s}^{\prime}$ on
$\mathsf{SW}$ which are not spin. Then there is an isometry $\varphi$ for
which $\varphi^{*}\mathfrak{s}=\mathfrak{s}^{\prime}$ if and only if
$Q_{\mathsf{SW}}(c_{1}(\mathfrak{s}),c_{1}(\mathfrak{s}))=Q_{\mathsf{SW}}(c_{1}(\mathfrak{s}^{\prime}),c_{1}(\mathfrak{s}^{\prime}))$

###### Proof.

This follows by the Witt extension theorem [45, Theorem $1.5.3$]: because
$Q_{\mathsf{SW}}$ is non degenerate, given two elements $x,y\neq 0$ satisfying
$Q(x)=Q(y)$, there is $A\in\mathrm{SO}(Q_{\mathsf{SW}})$ for which $Ax=y$. ∎

$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$e$$a$$b$$c$$d$$b$$c$$d$$e$$a$$e$$f$$a$$f$$b$$f$$c$$f$$d$$f$$c$$d$$e$$a$$b$$a$$b$$c$$d$$e$
Figure 2. A graphic depiction of the Seifert-Weber dodecahedral space. The
twelfth face is the one formed of the five external edges, and we consider it
to be on top of the others.

##  9\. 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

$\sum_{\ell(\gamma)\geq
R}\ell(\gamma_{0})\cdot\frac{2\sin(\mathrm{hol}(\gamma))}{|1-e^{\mathbb{C}l(\gamma)}|\cdot|1-e^{-\mathbb{C}l(\gamma)}|}\cdot\frac{1}{\ell}\cdot\left(-G(\ell/L)\right).$

for the signature case and

$\sum_{\ell(\gamma)\geq
R}\ell(\gamma_{0})\cdot\frac{2\sin(\mathrm{hol}(\tilde{\gamma}))\cdot{\cos(\varphi_{\tilde{\gamma}})}}{|1-e^{\mathbb{C}\ell(\gamma)}|\cdot|1-e^{-\mathbb{C}\ell(\gamma)}|}\cdot\left(-G(\ell/L)\right),$

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

$\sum_{\ell(\gamma)\in[T-1/2,T+1/2]}\ell(\gamma_{0}).$

To put this in context, the quantity

(18) $\sum_{\ell(\gamma)\leq T}\ell(\gamma_{0}).$

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

$\\#\\{\text{prime geodesics $\gamma$ with $\ell(\gamma)\leq T$}\\}\sim
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
$e^{\ell(\gamma)}$ is the analogue of $p$), the quantity (18) corresponds to
the Chebyshev function

$\psi(x)=\sum_{p^{k}\leq x}\log(p).$

It is well known [1, Chapter $4$] that the prime number theorem

$\\#\\{\text{prime numbers $p\leq x$}\\}\sim x/\log(x)$

is equivalent to the asymptotic

$\psi(x)\sim x.$

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
$\cosh(t)\mathbf{1}_{[-T,T]}$ (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 $\mathsf{SW}$ up to cutoff $R=8$ (rather than just the
injectivity radius), as in [29]. Furthermore, we have computed the spinc
length spectrum of $\mathsf{SW}$ for all spinc 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 $\mathsf{SW}$ has no small eigenvalues, and the
smallest parameter satisfies $r_{1}>2.8$, corresponding to $\lambda_{1}>8.84$.

######  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
$\mathrm{GL}_{2}.$ 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)444Note the very different scales of Figures 3
and 4.. In both cases, to exclude the parameter $t,$ we minimized the quantity

(19) $\sum_{n=1}^{\infty}\widehat{H}(r_{n})$

over the same space of functions used in [28], subject to the constraint
$\widehat{H}(t)=1,$ 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 $\lambda_{0}=0$), and is not included in the sum (19).

Figure 3. The picture for the imaginary parameters for the Laplacian on
functions on $\mathsf{SW}$. Figure 4. The picture for the real parameters for
the Laplacian on functions on $\mathsf{SW}$.

###  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

$\displaystyle H_{\nu}$
$\displaystyle=\left(\frac{1}{2}\mathbf{1}_{[-1,1]}\right)^{\ast
6}\cdot(e^{i\nu x}+e^{-i\nu x})$
$\displaystyle=2\cdot\left(\frac{1}{2}\mathbf{1}_{[-1,1]}\right)^{\ast
6}\cdot\cos(\nu x)$

which is the same kind of function we used in Section 5. In this case, we have

$\displaystyle-H_{\nu}^{{}^{\prime\prime}}(0)$
$\displaystyle=2*\left(\left(\left(\frac{1}{2}\mathbf{1}_{[-1,1]}\right)^{\ast
6}\right)^{{}^{\prime\prime}}(0)-\nu^{2}\cdot\left(\frac{1}{2}\mathbf{1}_{[-1,1]}\right)^{\ast
6}(0)\right)$ $\displaystyle=\frac{11}{20}\nu^{2}+\frac{1}{4},$

and

$\widehat{H_{\nu}}(t)=\left(\frac{\sin(t+\nu)}{t+\nu}\right)^{6}+\left(\frac{\sin(t-\nu)}{t-\nu}\right)^{6}.$

We will use the fact that that

$\left(\frac{\sin(x)}{x}\right)^{6}\geq 0.777\text{ for }x\in[-0.5,0.5].$

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 geodesics

$\displaystyle\left|\sum_{\gamma}\frac{\ell(\gamma_{0})}{|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|}H_{\nu}(\ell(\gamma))\right|$
$\displaystyle\leq\sum_{\gamma}\frac{\ell(\gamma_{0})}{|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|}H_{0}(\ell(\gamma))$
$\displaystyle\leq 4.827$

because $H_{0}$ is supported in $[-6,6]$ and we know the length spectrum up to
cutoff $R=8$. Furthermore, recalling that

$\displaystyle\lvert\sin(i+\nu)\lvert^{2}$
$\displaystyle=\cos^{2}(\nu)\sinh^{2}(1)+\sin^{2}(\nu)\cosh^{2}(1)$
$\displaystyle=\sinh^{2}(1)+\sin^{2}(\nu),$

we have that the contribution of the zero eigenvalue is bounded above by

(20) $2\cdot\left(\frac{\sinh(1)^{2}+\sin(\nu)^{2}}{1+\nu^{2}}\right)^{3}$

We are interested in upper bounds for the number of spectral parameters in
$[\nu-1/2,\nu+1/2]$ for $T\geq 3$. The quantity (20) is bounded above by
$0.0055$ for $\nu\geq 3$. Putting everything together, and recalling that
$\mathsf{SW}$ has volume about $11.19$, we get the following refined local
Weyl law.

######  Proposition 9.2.

For the eigenvalues of the Laplacian of functions on $\mathsf{SW}$, the upper
bound

$\\#\\{\text{parameters in }[\nu-1/2,\nu+1/2]\\}\leq 1.262\nu^{2}+6.793.$

holds for $\nu\geq 3$.

###  9.3. Bounds for closed geodesics

We will apply the trace formula of Theorem 3.1 to the function

$F_{c,T}=e^{-\frac{c(x-T)^{2}}{2}}+e^{-\frac{c(x+T)^{2}}{2}},$

which has Fourier transform

$\widehat{F}_{c,T}=\sqrt{\frac{2\pi}{c}}e^{-\frac{x^{2}}{2c}}\cdot(e^{iTx}+e^{-iTx})=2\sqrt{\frac{{2\pi}}{c}}e^{-\frac{x^{2}}{2c}}\cos(Tx).$

Here $c>0$ 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

$2\sqrt{\frac{2\pi}{c}}e^{1/{2c}}\cosh(T).$

Using Proposition 9.1, the contribution of the real parameters is bounded
above (independenty of $T$) by

$\displaystyle\sum_{r_{n}\text{ real}}\widehat{F}_{c,T}(r_{n})$
$\displaystyle=\sum_{n=3}^{\infty}\sum_{r_{n}\in[n-1/2,n+1/2]}\widehat{F}_{c,T}(r_{n})$
(21) $\displaystyle\leq\sum_{n=3}^{\infty}(1.262n^{2}+6.793)\cdot
2\sqrt{\frac{2\pi}{c}}e^{-\frac{(n-1/2)^{2}}{2c}}.$

The identity contribution is

(22)
$-\frac{\mathrm{vol}(Y)}{2\pi}F_{c,T}^{\prime\prime}(0)=\frac{\mathrm{vol}(Y)}{\pi}\cdot
e^{-\frac{cT^{2}}{2}}\cdot(c-c^{2}T^{2}).$

(this is negative provided $T^{2}>1/c$).

All these terms are explicitly computable, and we will use this information to
provide upper bounds on

$\sum_{l(\gamma)\in[T-1/2,T+1/2]}\ell(\gamma_{0}).$

as follows. Notice that

$\displaystyle|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|$
$\displaystyle\leq(e^{\ell}+1)(1+e^{-\ell})$ $\displaystyle=2\cosh(\ell)+2$

and

$F_{c,T}(\ell)\geq e^{-c/8}\text{ for }\ell\in[T-1/2,T+1/2]$

We therefore have

$\displaystyle\sum_{\gamma}\frac{\ell(\gamma_{0})}{|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|}F_{c,T}(\ell(\gamma))$
$\displaystyle\geq$
$\displaystyle\sum_{\ell(\gamma)\in[T-1/2,T+1/2]}\frac{\ell(\gamma_{0})}{2\cosh(T+1/2)+2}e^{-c/8}$
$\displaystyle=$
$\displaystyle\frac{e^{-c/8}}{2\cosh(T+1/2)+2}\cdot\sum_{\ell(\gamma)\in[T-1/2,T+1/2]}{\ell(\gamma_{0})}.$

and we can use the explicit upper bound for the first expression, obtained by
combining the trace formula with the estimates (9.3) and (22) , to get an
upper bound for $\sum_{\ell(\gamma)\in[T-1/2,T+1/2]}{\ell(\gamma_{0})}$,
depending on a parameter $c$. We see empirically that we obtain the best
estimate for $c=5$, and we have the following:

######  Proposition 9.3.

For $T\geq 7.5$, we have

$\sum_{\ell(\gamma)\in[T-1/2,T+1/2]}{\ell(\gamma_{0})}\leq A(T),$

where $A(T)$ is an explicit expression in $T$ readily obtained from the
discussion above.

We do not write explicitly the explicit expression of $A(T)$ as it is quite
long and not particularly illuminating, but we remark that the leading term is
of the form $C\cdot e^{2T}$.

###  9.4. The geometric error for $\eta$

Finally, we will conclude by bounding the error in the evaluation of the eta
invariant using the standard Gaussian $G(x)=e^{-x^{2}/2}$ in the formula (14),
and evaluating the geometric side up to length $R=7.5$. We have
$\widehat{G}(t)=\sqrt{2\pi}e^{-t^{2}/2}$, and

$\int_{0}^{\infty}\widehat{G}(t)dt=\pi,$

these errors correspond to

$\frac{2}{\pi}\sum_{\ell(\gamma)\geq
7.5}\ell(\gamma_{0})\cdot\frac{\sin(\mathrm{hol}(\gamma))}{|1-e^{\mathbb{C}l(\gamma)}|\cdot|1-e^{\mathbb{C}l(\gamma)}|}\cdot\frac{1}{\ell(\gamma)}\cdot\left(-G(\ell/L)\right)$

for the signature case and

$\frac{2}{\pi}\sum_{\ell(\gamma)\geq
7.5}\ell(\gamma_{0})\cdot\frac{\sin(\mathrm{hol}(\tilde{\gamma}))\cdot{\cos}(\varphi_{\tilde{\gamma}})}{|1-e^{\mathbb{C}\ell(\gamma)}|\cdot|1-e^{-\mathbb{C}\ell(\gamma)}|}\cdot\frac{1}{\ell(\gamma)}\cdot\left(-G(\ell/L)\right)$

for the Dirac operator; again, $L$ is the point at which we split the
integral. We can therefore bound the error in the geometric side in all cases
by

$\frac{2}{\pi}\sum_{\ell(\gamma)\geq
7.5}\frac{\ell(\gamma_{0})}{|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|}\frac{G(\ell(\gamma)/L)}{\ell(\gamma)}.$

We can break the sum in two parts, one taking into account
$\ell(\gamma)\in[7.5,8]$ and one for $\ell(\gamma)\geq 8$. The first part can
be computed explicitly because we know the (standard) length spectrum of
$\mathsf{SW}$ up to cutoff $R=8$. For the second part, noticing that

$|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|\geq(e^{\ell}-1)(1-e^{-\ell})=2\mathrm{sinh}(\ell)-2,$

we have

$\displaystyle\frac{2}{\pi}\sum_{\ell(\gamma)\geq
8}\frac{\ell(\gamma_{0})}{|1-e^{\mathbb{C}\ell(\gamma)}||1-e^{-\mathbb{C}\ell(\gamma)}|}\frac{G(\ell(\gamma)/L)}{\ell(\gamma)}$
$\displaystyle\leq\frac{1}{\pi}\sum_{n=8}^{\infty}\frac{e^{-n^{2}/2L^{2}}}{(\sinh(n)-1)\cdot
n}\cdot\sum_{\ell(\gamma)\in[n,n+1]}\ell(\gamma_{0})$
$\displaystyle=\frac{1}{\pi}\sum_{n=8}^{\infty}\frac{e^{-n^{2}/2L^{2}}}{(\sinh(n)-1)\cdot
n}\cdot A(n+1/2)$

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 $\eta$ invariant for either the signature or the Dirac
operator using a standard Gaussian $G(x)=e^{-x^{2}/2}$, 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 $0.0376$.555For the evaluation of this
sum, we used the fact that the general term decays extremely rapidly, as
$A(n)=O(e^{2n})$; for example, the term $n=31$ is of the order of $10^{-57}$.

##  10\. 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.

###  10.1. The signature eta invariant.

We have the following.

######  Proposition 10.1.

The signature eta invariant of $\mathsf{SW}$ is
$\eta_{\mathrm{sign}}=1.31111111\dots$

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 $\mathsf{SW}$ to be

$\mathrm{cs}=-0.033333\dots$

The relation

$3\eta_{\mathrm{sign}}=2\mathrm{cs}+\tau\mod 2\mathbb{Z}$

holds [3], where $\tau$, which is the number of $2$-primary summands in
$H_{1}(\mathsf{SW},\mathbb{Z})$, is zero in our case. Thus

(23) $\eta_{\mathrm{sign}}=-0.022222\dots+\frac{2}{3}n$

for some $n\in\mathbb{Z}$. Therefore, approximating the eta invariant to
within error less than $\frac{1}{3}$ pins down its value to high accuracy.
Approximating $\eta_{\mathrm{sign}}$ well is straightforward because we have a
good understanding of the small coexact 1-form eigenvalues on $\mathrm{SW}$
(see Figure 5, which we used in [29] to show $\lambda_{1}^{*}>2$):

  * •

The smallest eigenspace corresponds to

(24) $\sqrt{\lambda_{1}^{*}}\in[1.42787720680237,1.43033743858337]$

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 $G(x)=e^{-x^{2}/2}$ 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 $\widetilde{\sqrt{\lambda_{1}^{*}}}$ of the interval in
(24), and consider the remaining part of the spectral sum as an error term.

As $\int_{0}^{\infty}\widehat{G}(t)dt=\pi$, we have the approximation

$\displaystyle\eta_{\mathrm{sign}}$
$\displaystyle\approx\frac{1}{\pi}\left(6\cdot\int_{1.7\cdot\widetilde{\sqrt{\lambda_{1}^{*}}}}^{\infty}\sqrt{2\pi}e^{-t^{2}/2}dt-\sum_{\ell(\gamma)\leq
7.5}\ell(\gamma_{0})\cdot\frac{2\sin(\mathrm{hol}(\gamma))}{|1-e^{\mathbb{C}l(\gamma)}|\cdot|1-e^{-\mathbb{C}l(\gamma)}}\cdot\frac{1}{\ell(\gamma)}\cdot
e^{-\frac{\ell^{2}}{2}\cdot(1.7)^{2}}\right)$
$\displaystyle=1.31102382358\dots$

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

$\frac{6}{\pi}\cdot\int_{1.7\times 1.427877}^{1.7\times
1.4303375}\sqrt{2\pi}e^{-t^{2}/2}dt\leq 0.00105.$

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

$\cos(Tx)\cdot\left(\frac{1}{2\delta}\mathbf{1}_{[-\delta,\delta]}\right)^{\ast
6}\text{ where }\delta=8/6$

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 $[2,3]$, and each contributes at
most $\frac{1}{\pi}\int_{1.7\times
2}^{\infty}\sqrt{2\pi}e^{-t^{2}/2}dt<0.00068.$

  * •

there are at most $21$ spectral parameters in $[3,4]$, and each contributes at
most $\frac{1}{\pi}\int_{1.7\times
3}^{\infty}\sqrt{2\pi}e^{-t^{2}/2}dt<3.4\times 10^{-7}$

  * •

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 $J_{\mathrm{SW}}(t)$ for coexact $1$-forms, computed
with cutoff $R=8$, cf. [29].

######  Remark 10.1.

As the numerical result suggests, $\eta_{\mathrm{sign}}$ is a rational number.
This follows because the invariant trace field
$\mathbb{Q}(\sqrt{-1-2\sqrt{5}})$ of $\mathsf{SW}$ is $\mathrm{CM}$-embedded,
see [39]. In fact, one can show that

$\eta_{\mathrm{sign}}=59/45$

using the multiplicativity of the Chern-Simons invariants under covers, and
the fact that $\mathsf{SW}$ admits a $60$-fold cover with an orientation
reversing isometry (for which $\eta_{\mathrm{sign}}$ is therefore $0$).

###  10.2. The spin structure

We now focus on the computation of $h(\mathsf{SW},\mathfrak{s}_{0})$ where
$\mathfrak{s}_{0}$ is the unique spin structure on $\mathrm{SW}.$ This is the
simplest spinc structure to handle because

$-2h(\mathsf{SW},\mathfrak{s}_{0})=\eta_{\mathrm{sign}}/4+\eta_{\mathrm{Dir}}=n/4$

for some $n\in\mathbb{Z}$. To see this, recall the classical theorem that
every spin three-manifold bounds a compact spin four-manifold
$(X,\mathfrak{s}_{X})$; 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

$\displaystyle\eta_{\mathrm{sign}}/4+\eta_{\mathrm{Dir}}$
$\displaystyle=\frac{1}{4}\left(c_{1}(\mathfrak{s}_{X})^{2}-2\chi(X)-3\sigma(X)-2\right)\mod\mathbb{Z}$