Nonacyclic Reidemeister torsions of manifolds of odd dimension

Takefumi Nosaka 111 E-mail address<EMAIL_ADDRESS>Koki Yanagida
222 E-mail address<EMAIL_ADDRESS>Naoko Wakijo 333 E-mail
address<EMAIL_ADDRESS>
###### Abstract

Given an oriented closed manifold $M$ of odd dimension and a unitary
representation $\rho:\pi_{1}(M)\rightarrow{\rm GL}_{n}({\mathbb{F}})$, we
define a Reidemeister torsion, even if the cohomology associated with $\rho$
is not acyclic. As corollaries, we introduce some topological invariants of
$M$, which include the nonacyclic extensions of abelian torsions and the
Alexander polynomials of links. Further, we propose a volume form of the ${\rm
SU}(n)$-character varieties of $M$. Moreover, we compute the Reidemeister
torsions of some representations of 3-manifolds and compare the works of
Farber–Turaev.

Keywords
Reidemeister torsion, 3-manifolds, Poincaré duality

Subject Codes
19J10, 58K65, 57P10, 15A63

##  1 Introduction

Let ${\mathbb{F}}$ be a field with involution
$\bar{}:{\mathbb{F}}\rightarrow{\mathbb{F}}$, and let $C^{*}$ be a bounded
chain complex consisting of finite dimensional ${\mathbb{F}}$-vector spaces.
Roughly speaking, a combinatorial (Reidemeister)-torsion of $C^{*}$ is an
alternating product of the determinants of $C^{i}$’s (see §2 for the
definition). For a closed oriented manifold $X$, if the cellular complex
$C^{*}_{\rho}(X)$ of $X$ with local coefficients over ${\mathbb{F}}$ is
acyclic and satisfies certain conditions, the torsion of $C^{*}_{\rho}(X)$ is
a topological invariant and studied from many aspects; e.g., see [Tur, Nic]
for its applications and developments. Meanwhile, if $C_{*}^{\rho}(X)$ is not
acyclic with the choice of the basis of cohomology $H^{*}_{\rho}(X)$, we can
define a (non-acyclic) torsion $\in{\mathbb{F}}$ that depends on the choice of
the basis. By the parity of Poincaré duality, the existing approaches of the
nonacyclic torsion are different according to the parity of
$\mathop{\mathrm{dim}}\nolimits(X)$; see [Söz, Wit] and the references therein
for even cases of $\mathop{\mathrm{dim}}\nolimits(X)$, and see [PY] for knot
cases. We focus on nonacyclic torsions with $\mathop{\mathrm{dim}}\nolimits
X=2m+1$ for some $m\in{\mathbb{N}}$. Inspired by analytic approaches to
torsions in the Ray–Singer norm, Farber and Turaev [Far, FT] define a torsion
on the determinant line of the cohomology of any flat vector bundle over $X$,
and a Poincaré-Reidemeister scalar product from their torsion; see also
Appendix B. In the definition, they assume ${\mathbb{F}}={\mathbb{R}}$ or
${\mathbb{F}}={\mathbb{C}}$ and use the duality from the twisted intersection
forms on homology. However, the definition of the twisted intersection form
has ambiguous aspects, and the determinant line of the cohomology is not
investigated as a quantitative object; furthermore, few examples of computing
the scalar products are provided.

In this paper, from a cohomological viewpoint with a Poincaré duality, we will
define combinatorially a nonacyclic torsion with respect to representation
$\rho:\pi_{1}(X)\rightarrow{\rm GL}_{n}({\mathbb{F}})$ with
$\mathop{\mathrm{dim}}\nolimits X=2m+1$, where $\rho$ is required to admit a
bilinear form satisfying a condition; see Theorem 3.1. The definition
indicates that ${\mathbb{F}}$ may be a field of characteristic zero, and the
torsion is independent of the choice of the basis of $H^{*}_{\rho}(X)$; see
Theorem 3.1. Further, we show (Appendix B) that torsion is a (square-root)
modification of the Poincaré-Reidemeister scalar product. However, because
nonacyclic torsion is defined in a quotient group of the form
${\mathbb{F}}^{\times}/\\{x\bar{x}\mathrm{det}(\rho(y))\mid
x\in{\mathbb{F}}^{\times},y\in\pi_{1}(X)\\}$, the quotient group may be
trivial in some cases; see Section 5 for some computations of the group. To
obtain nontrivial torsions, we restrict ourselves to cases where the quotient
group is not trivial.

In such nontrivial cases, we suggest applications from the non-acyclic
torsions some topological invariants of a closed manifold $X$ of odd
dimension; see Section 6. When $\rho$ is trivial, we show the relationships
between the torsion and ordinary cohomology; see Sections 6.1 and 6.2. For
nontrivial cases of $\rho$, we see that the abovementioned torsion produces
abelian Reidemeister torsions in nonacyclic cases (Section 6.3). As an
application, we propose a generalization of the Alexander polynomial of links
in Section 6.4. As another application, we ensure that the torsion generates a
volume form on the ${\rm SU}(n)$-character variety of $X$, as an analogy to
the volume form of the flat moduli spaces of closed surfaces [Wit] (see
Section 4).

Finally, in Section 8, we focus on the case of $\mathop{\mathrm{dim}}\nolimits
X=3$ and provide examples of computing the torsions of 3-manifolds. In the
three-dimensional case, we can compute the Poincaré duality because some
procedures of computing the cup product are known [S86, Tro] ; in doing so, we
obtain nontrivial torsions in some cases. In addition, we argue further
refinements of the torsions; see Propositions 7.2 amd 7.3. Although this paper
suggests quantitative definitions and research of nonacyclic torsions, their
applications and interesting properties need to be developed in future
studies.

Conventional terminology. Throughout this paper, ${\mathbb{F}}$ and
${\mathbb{F}}^{\times}={\mathbb{F}}\setminus\\{0\\}$ denote a commutative
field and the multiplicative group, respectively. Further, we suppose an
involution $\bar{}:{\mathbb{F}}\rightarrow{\mathbb{F}}$ (possibly,
$\bar{}=\mathrm{id}_{{\mathbb{F}}}$). $X$ represents an oriented closed
connected $C^{\infty}$-manifold of dimension $2m+1$.

##  2 Review: Refined Reidemeister torsions of ${\rm GL}_{n}$-representations

Let us briefly review algebraic torsions. Let $V$ be a $d$-dimensional vector
space over a field ${\mathbb{F}}$. For two bases of $V$, as in
${\mathbf{b}}=\\{b_{1},\ldots,b_{d}\\},{\mathbf{c}}=\\{c_{1},\ldots,c_{d}\\}\subset
V$, we obtain a transition matrix $P\in{\rm GL}(d;{\mathbb{F}})$ that
satisfies $b_{i}=\sum_{j=1}^{d}P_{(j,i)}c_{j}$, where $P_{(j,i)}$ represents
the $(j,i)$-th entry of $P$. We write
$[{\mathbf{b}}/{\mathbf{c}}]\in{\mathbb{F}}^{\times}$ for the determinant of
$P$. Consider a cochain complex

$C^{*}:0\longrightarrow
C^{0}\overset{\partial_{0}}{\longrightarrow}C^{1}\overset{\partial_{1}}{\longrightarrow}\cdots\xrightarrow{\
\partial_{m-2}\ }C^{m-1}\xrightarrow{\ \partial_{m-1}\ }C^{m}\longrightarrow
0,$ (1)

such that each $C^{i}$ is a finite-dimensional vector space over
${\mathbb{F}}$, and we choose a basis ${\mathbf{c}}_{i}$ of $C^{i}$ for each
$i$. We employ the common notations
$Z^{i}=\mathop{\mathrm{Ker}}\nolimits(\partial_{i})$,
$B^{i}=\mathop{\mathrm{Im}}\nolimits(\partial_{i-1})$, and
$H^{i}=Z^{i}/B^{i}$. Considering the two exact sequences

$0\longrightarrow Z^{i}\longrightarrow
C^{i}\overset{\partial_{i}}{\longrightarrow}B^{i+1}\longrightarrow
0,\quad\quad 0\longrightarrow B^{i}\longrightarrow
Z^{i}{\longrightarrow}H^{i}\longrightarrow 0$

over ${\mathbb{F}}$, we choose sections $\mathfrak{s}_{i}:B^{i+1}\rightarrow
C^{i}$ and $\mathfrak{s}_{i}^{\prime}:H^{i}\rightarrow Z^{i}$. For $i\leq m$,
we choose a basis ${\mathbf{b}}_{i}$ of $B^{i}$, and we fix its lift to
$C_{i+1}$; let $\widetilde{{\mathbf{b}}}_{i}$ denote the lift, that is,
$\widetilde{{\mathbf{b}}}_{i+1}=\mathfrak{s}_{i}({\mathbf{b}}_{i+1})$.
Similarly, we select a basis ${\mathbf{h}}_{i}$ of $H^{i}$ and write
$\widetilde{{\mathbf{h}}}_{i}$ for its lift in $Z^{i}$. Then, under the
identifications, the (ordered) union
$({\mathbf{b}}_{i}\cup\widetilde{{\mathbf{h}}}_{i}\cup\widetilde{{\mathbf{b}}}_{i+1})$
is a basis of $C^{i}$. Let ${\mathbf{c}}$ and ${\mathbf{h}}$ be the unions of
the basis
$({\mathbf{c}}_{0}\cup{\mathbf{c}}_{1}\cup\cdots\cup{\mathbf{c}}_{m})$ and
$({\mathbf{h}}_{0}\cup{\mathbf{h}}_{1}\cup\cdots\cup{\mathbf{h}}_{m})$,
respectively. The torsion of $C^{*}$ is defined as

$\tau(C^{*},{\mathbf{c}},{\mathbf{h}}):=\prod_{i=0}^{m}[({\mathbf{b}}_{i}\cup\widetilde{{\mathbf{h}}}_{i}\cup\widetilde{{\mathbf{b}}}_{i+1})/{\mathbf{c}}_{i}]^{(-1)^{i+1}}\in{\mathbb{F}}^{\times}.$

It is known (see [Tur]) that $\tau(C^{*},{\mathbf{c}},{\mathbf{h}})$ is
independent of the choices of the basis
$({\mathbf{b}}_{0},{\mathbf{b}}_{1},\ldots,{\mathbf{b}}_{m})$ and the sections
$\mathfrak{s}_{i}$ and $\mathfrak{s}_{i}^{\prime}$. However, the torsion
depends on the choices of $\mathbf{c}_{i}$ and $\mathbf{h}_{i}$. More
precisely, if we select other such bases, $\mathbf{c}_{i}^{\prime}$ and
$\mathbf{h}_{i}^{\prime}$, the following holds [Tur, Remark 1.4.1]:

$\tau(C^{*},{\mathbf{c}},{\mathbf{h}})=\tau(C^{*},{\mathbf{c}}^{\prime},{\mathbf{h}}^{\prime})\prod_{i=0}^{m}([{\mathbf{c}}_{i}/{\mathbf{c}}^{\prime}_{i}][{\mathbf{h}}_{i}^{\prime}/{\mathbf{h}}_{i}])^{(-1)^{i+1}}\in\mathbb{F}^{\times}.$
(2)

Next, let us review Reidemeister torsions. Let $Y$ be a connected finite CW-
complex. Take a ${\rm GL}_{n}$-representation $\rho:\pi_{1}(Y)\rightarrow{\rm
GL}_{n}(\mathbb{F})$, and consider ${\mathbb{F}}^{n}$ as a left
${\mathbb{Z}}[\pi_{1}(Y)]$-module. Let $\widetilde{Y}$ be the universal
covering space of $Y$ as a CW complex and $C_{*}(\widetilde{Y};{\mathbb{Z}})$
be the cellular complex associated with the CW complex. This
$C_{*}(\widetilde{Y};{\mathbb{Z}})$ can be viewed as a left free
${\mathbb{Z}}[\pi_{1}(Y)]$-module by covering transformations. Let us choose a
basis ${\mathbf{c}}_{i}$ of the free ${\mathbb{Z}}[\pi_{1}(Y)]$-module
$C_{i}(\widetilde{Y};{\mathbb{Z}})$. Then, we have the chain complex of the
local system
$C_{*}(\widetilde{Y};{\mathbb{Z}})\otimes_{\pi_{1}(Y)}\mathbb{F}^{n}$. Dually,
the cochain complex of the local system is defined by

$C^{*}_{\rho}(Y;\mathbb{F}^{n}):=\mathrm{Hom}_{\mathbb{Z}[\pi_{1}(Y)]\textrm{-mod}}(C_{*}(\widetilde{Y};{\mathbb{Z}}),\mathbb{F}^{n}).$

As a special case, if $n=1$ and $\rho$ is trivial, we use the notation $C^{*}$
instead of $C_{\rho}^{*}$. For a standard basis
${\mathbf{f}}\subset{\mathbb{F}}^{n}$, the tensor
${\mathbf{c}}_{i}\otimes{\mathbf{f}}$ is a basis of the vector space
$C_{i}(\widetilde{Y};{\mathbb{Z}})\otimes_{\pi_{1}(Y)}\mathbb{F}^{n}$. Let
${\mathbf{c}}_{i}^{*}$ be the dual basis of $C^{i}_{\rho}(Y;\mathbb{F}^{n})$.
Let ${\mathbf{c}}_{Y}$ denote the union
${\mathbf{c}}_{0}^{*}\cup{\mathbf{c}}_{1}^{*}\cup\cdots\cup{\mathbf{c}}_{{\rm
dim}Y}^{*}$. Now, let us consider the subgroups

$\mathrm{det}(\rho(\pi_{1}(Y))):=\\{\mathrm{det}(\rho(x))\mid
x\in\pi_{1}(Y)\\}\subset{\mathbb{F}}^{\times},$
$\pm\mathrm{det}(\rho(\pi_{1}(Y))):=\\{\varepsilon\mathrm{det}(\rho(x))\mid
x\in\pi_{1}(Y),\varepsilon\in\\{\pm 1\\}\\}\subset{\mathbb{F}}^{\times},$

and the quotient group
$\mathbb{F}^{\times}/\pm\mathrm{det}(\rho(\pi_{1}(Y)))$. Furthermore, with a
choice of basis $\mathbf{h}_{i}$ of the cohomology
$H^{i}_{\rho}(Y;\mathbb{F}^{n})$, the Reidemeister torsion of ($Y,\rho$) is
defined as

$\tau(C^{*}_{\rho}(Y;\mathbb{F}^{n}),{\mathbf{c}}_{Y},{\mathbf{h}})\in\mathbb{F}^{\times}/\pm\mathrm{det}(\rho(\pi_{1}(Y)))$

modulo $\pm\mathrm{det}(\rho(\pi_{1}(Y)))$. As is well known [Mil, Tur], the
torsion modulo signs does not depend on the choice of ${\mathbf{c}}_{Y}$.

In addition, let us review the sign-refined torsions by Turaev [Tur, Dub], and
state Theorem 2.1. Let $H^{*}(Y;{\mathbb{R}})$ be the cohomology over
${\mathbb{R}}$. Assuming an orientation of $\oplus_{i\geq
0}H^{i}(Y;{\mathbb{R}})$, we choose a basis
$\mathbf{h}_{i}^{{\mathbb{R}}}\subset H^{i}(Y;{\mathbb{R}})$ so that the
sequence $(\mathbf{h}_{0}^{{\mathbb{R}}},\mathbf{h}_{1}^{{\mathbb{R}}},\dots)$
is a positive basis in the oriented vector space $H^{*}(Y;{\mathbb{R}})$.
Consider

$\tilde{\tau}(C^{*}(Y;\mathbb{R}),\mathbf{c}_{Y},\mathbf{h}^{{\mathbb{R}}}):=(-1)^{N(Y)}\tau(C^{*}(Y;\mathbb{R}),\mathbf{c}_{Y},\mathbf{h}^{{\mathbb{R}}})\in{\mathbb{R}}^{\times},$

where

$N(Y)=\sum_{i=0}^{{\rm dim}(Y)}\bigl{(}\sum_{j=0}^{i}\mathrm{dim}H^{{\rm
dim}(Y)-j}(Y;{\mathbb{R}})\sum_{j=0}^{i}\mathrm{dim}C^{{\rm
dim}(Y)-j}(Y;{\mathbb{R}})\bigr{)}\in{\mathbb{Z}}/2{\mathbb{Z}}.$ (3)

Then, the refined torsion can be defined as

$\tau^{0}_{\rho}(Y,\mathbf{h}):=\mathrm{sign}\bigl{(}\tilde{\tau}(C^{*}(Y;\mathbb{R}),\mathbf{c}_{Y},\mathbf{h}^{{\mathbb{R}}})\bigr{)}^{n}\cdot\mathcal{T}(C^{*}_{\rho}(Y;\mathbb{F}^{n}),\mathbf{c}_{Y},\mathbf{h})\in{\mathbb{F}}^{\times}.$
(4)

If $n$ is even, then the sign is +1, that is,
$\tau^{0}_{\rho}(Y,\mathbf{h})=\tau(C^{*}_{\rho}(Y;\mathbb{F}^{n}),\mathbf{c}_{Y},\mathbf{h})$.
Then, the topological invariance is given as

######  Theorem 2.1 (see [Tur, Chapter 18] or [Dub, Chapter 2]).

The sign-refined torsion
$\tau^{0}_{\rho}(Y,\mathbf{h})\in{\mathbb{F}}^{\times}/\mathop{\mathrm{det}}\nolimits(\pi_{1}(Y))$
is independent of the order of the oriented cells of $Y$, and the choice of
$\mathbf{h}^{{\mathbb{R}}}$; however, it does depend on the choice of
$\mathbf{h}$. Furthermore, the torsion is invariant under simple homotopy
equivalences that preserve the homology orientation.

As a corollary, let us consider the case where $Y$ is an oriented closed
$C^{\infty}$-manifold. Recall that any two triangulations of an oriented
closed $C^{\infty}$-manifold $N$ are simple homotopy equivalent (see, e.g.,
[Tur, §II.8]); consequently, if $Y$ is a triangulation of $N$, the sign-
refined torsion yields a topological invariant of $N$ associated with
$\rho:\pi_{1}(N)\rightarrow{\rm GL}_{n}({\mathbb{F}})$ and $\mathbf{h}$.

##  3 Nonacyclic torsions of the closed manifolds of odd dimension

Theorem 2.1 and (2) imply that the refined torsion (4) depends on
$\mathbf{h}$, and therefore, it is reasonable to find some torsions that
depend only on the representation $\rho:\pi_{1}(Y)\rightarrow{\rm
GL}_{n}({\mathbb{F}})$. Previous studies have investigated torsions in the
case where $Y$ is an oriented closed manifold of even dimension, see, e.g.,
[Söz, Wit]. In contrast, this paper addresses nontrivial torsions in the case
$\mathrm{dim}X=2m+1$ in terms of Poincaré duality.

To this end, we prepare some cup products as follows. Let us fix a bilinear
form $\psi:{\mathbb{F}}^{n}\times{\mathbb{F}}^{n}\rightarrow{\mathbb{F}}$
satisfying the $\rho$-invariance, that is, $\psi(\rho(g)\cdot v,\rho(g)\cdot
w)=\psi(v,w)$ hold for any $g\in\pi_{1}(X),v,w\in{\mathbb{F}}^{n}$.
Furthermore, suppose that $\psi$ is either Hermitian or anti-Hermitian. Using
the cup product $\smile$, consider the composite map

$H^{i}_{\rho}(X;{\mathbb{F}}^{n})\otimes
H^{2m+1-i}_{\rho}(X;{\mathbb{F}}^{n})\stackrel{{\scriptstyle\smile}}{{\longrightarrow}}H^{2m+1}_{\rho}(X;{\mathbb{F}}^{n}\otimes{\mathbb{F}}^{n})\xrightarrow{\
\cap[X]\
}H^{0}(X;{\mathbb{F}}^{n}\otimes{\mathbb{F}}^{n})\stackrel{{\scriptstyle\psi_{*}}}{{\longrightarrow}}{\mathbb{F}},$
(5)

where $\cap[X]$ represents the cap-product with the fundamental class $[X]\in
H_{2m+1}(X;{\mathbb{Z}})$, and the last map $\psi_{*}$ represents the coupling
with $\psi$. Further, we define $N({\mathbb{F}})$ to be $\\{y\bar{y}\mid
y\in{\mathbb{F}}^{\times}\\}$ as a multiplicative subgroup of
${\mathbb{F}}^{\times}$, and we consider the quotient group

${\mathbb{F}}^{\times}/\langle
N({\mathbb{F}}),\mathop{\mathrm{det}}\nolimits(\rho(\pi_{1}(X)))\rangle={\mathbb{F}}^{\times}/\\{x\bar{x}\mathrm{det}(\rho(g))\mid
x\in{\mathbb{F}}^{\times},g\in\pi_{1}(X)\\}.$

Then, the topological invariance of the refined torsion is obtained as
follows:

######  Theorem 3.1.

Let $\psi:{\mathbb{F}}^{n}\times{\mathbb{F}}^{n}\rightarrow{\mathbb{F}}$ be an
(anti-)Hermitian $\rho$-invariant bilinear form. We consider the basis
${\mathbf{h}}_{j}$ of $H^{j}(X;{\mathbb{F}}^{n})$ for $j\leq m.$

Assuming the nondegeneracy of (5), which defines uniquely
${\mathbf{h}}_{j}^{\rm dual}\subset H^{2m+1-j}(X;{\mathbb{F}}^{n})$ as the
dual basis of ${\mathbf{h}}_{j}$, the refined torsion is

$\tau^{0}_{\rho}(X,\\{\mathbf{h}_{0},\mathbf{h}_{1},\dots,{\mathbf{h}}_{m},\mathbf{h}_{0}^{\rm
dual},\mathbf{h}_{1}^{\rm dual},\dots,{\mathbf{h}}_{m}^{\rm
dual}\\})\in{\mathbb{F}}^{\times}/\langle
N({\mathbb{F}}),\mathop{\mathrm{det}}\nolimits(\rho(\pi_{1}(X))\rangle$ (6)

modulo $N({\mathbb{F}})$ and $\mathop{\mathrm{det}}\nolimits(\rho(\pi_{1}(X))$
is independent of the choice of the basis
${\mathbf{h}}_{0},\dots,{\mathbf{h}}_{m}$.

###### Proof.

We can show only that the torsion is independent of the choice of
${\mathbf{h}}_{j}$ and the order of ${\mathbf{h}}_{j}$. The independence of
the order is obvious because any change in the order of ${\mathbf{h}}_{j}$ is
parallel to that of ${\mathbf{h}}_{j}^{\rm dual}$. To show the remaining
independence of ${\mathbf{h}}_{j}$, we choose another ordered basis
${\mathbf{h}}_{j}^{\prime}$ of $H^{j}(X;{\mathbb{F}}^{n})$. Then, we can
easily check
$[{\mathbf{h}}_{j}/{\mathbf{h}}_{j}^{\prime}]=\overline{[{\mathbf{h}}_{j}^{\rm
dual}/({\mathbf{h}}_{j}^{\prime})^{\rm dual}]}^{-1}$. Thus, by (2), the
torsion (6) is equal to

$\tau^{0}_{\rho}(X,\\{\mathbf{h}_{0}^{\prime},\dots,{\mathbf{h}}_{m}^{\prime},(\mathbf{h}_{0}^{\prime})^{\rm
dual},\dots,({\mathbf{h}}_{m}^{\prime})^{\rm
dual}\\})\prod_{j=0}^{m}([{\mathbf{h}}_{j}/{\mathbf{h}}_{j}^{\prime}]\overline{[{\mathbf{h}}_{j}/{\mathbf{h}}_{j}^{\prime}]})^{(-1)^{j+1}}\in{\mathbb{F}}^{\times}.$
(7)

Thus, the torsion in the quotient group
${\mathbb{F}}^{\times}/N({\mathbb{F}})$ is independent of the choice of the
base ${\mathbf{h}}_{0},\dots,{\mathbf{h}}_{m}$ as required. ∎

To obtain the refined torsion, it is sensible to find a condition that ensures
nondegeneracy (5). Such a condition is obtained as follows:

######  Proposition 3.2.

Let $\psi:{\mathbb{F}}^{n}\times{\mathbb{F}}^{n}\rightarrow{\mathbb{F}}$ be a
$\rho$-invariant bilinear form. If $\psi$ is nondegenerate and field
${\mathbb{F}}$ is of characteristic zero, then there is an isomorphism
$D:H^{i}_{\rho}(X;{\mathbb{F}}^{n})\cong
H_{\rho}^{2m+1-i}(X;{\mathbb{F}}^{n})$ and the bilinear form (5) is
nondegenerate.

Since this proposition may be known (see, e.g., [HSW]), we defer the proof
into Appendix A. Here, it is worth noting that $D(ax)=\bar{a}D(x)$ for
$a\in{\mathbb{F}},x\in H^{i}_{\rho}(X;{\mathbb{F}}^{n}).$ We also show that
the refined torsion in Theorem 3.1 is a modification of the Poincaré-
Reidemeister (scalar) product [Far, FT]; see Appendix B for details.

##  4 Torsions as volume forms on ${\rm SU}(n)$-character varieties

Witten [Wit] reported that the adjoint torsion of any closed surface $\Sigma$
yields a volume form on the conjugacy classes set of
$\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(\Sigma),G)$ where $G$ is a semisimple
Lie group; see also [Söz] for higher dimensional cases of
$\mathop{\mathrm{dim}}\nolimits X/2\in{\mathbb{Z}}$.

As an application of our torsions, if $\mathop{\mathrm{dim}}\nolimits X=3$, we
analogously propose a volume form of the conjugacy classes
$\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(X),G)/$ conj using the torsion (6)
although we restrict the special unitary representations.

Let ${\rm SU}(n)$ be the special unitary group as a subgroup of ${\rm
GL}_{n}({\mathbb{C}})$. We fix the Hermitian metric on ${\mathbb{C}}$, which
takes $(v,w)$ to ${}^{t}\bar{v}w$. Then, we fix a lie group $G$ and an ${\rm
SU}$-representation $\rho^{\prime}:G\rightarrow{\rm SU}(n)$. The unitary group
ensures a Hermitian non-degenerate
$\psi:{\mathbb{C}}^{n}\times{\mathbb{C}}^{n}\rightarrow{\mathbb{C}}$
satisfying the condition in Proposition 3.2. For example, if $G$ is compact,
any representation over ${\mathbb{C}}$ is unitary. A canonical isomorphism
${\mathbb{C}}^{\times}/N({\mathbb{C}})\cong S^{1}$ will be presented in
Example 5.1. Using Theorem 3.1, for any homomorphism $f:\pi_{1}(X)\rightarrow
G$, we can define the torsion $\tau^{0}_{\rho^{\prime}\circ f}(X)$ in (6),
which lies in $\\{x\in{\mathbb{C}}\mid x\bar{x}=1\\}$. If two homomorphisms
$f$ and $f^{\prime}$ are conjugate, the resulting torsions are equal by
definitions. Thus, we obtain a map

$\frac{\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(X),G)}{\mathrm{conjugate}}\longrightarrow
U(1)=\\{x\in{\mathbb{C}}\mid
x\bar{x}=1\\};f\longmapsto\tau^{0}_{\rho^{\prime}\circ f}(X).$ (8)

If $G={\rm SU}(n)$, the domain is denoted by $R_{G}(X)$ and sometimes called
the ${\rm SU}(n)$-character variety (of $M$). Here, the topology on $R_{G}(X)$
is a quotient topology from $\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(X),G)$. As
an example, we compute the torsions of some Seifert 3-manifolds, when $G={\rm
SU}(2)$; see §8.2. Finally, we pose a problem.

######  Problem 4.1.

Is the torsion function (8) continuous? Moreover, if $G$ is semi-simple and
contained in ${\rm SU}(m)$ and $\rho^{\prime}$ is the adjoint representation,
then is the torsion function locally constant?

In contrast, we remark on known results: first, if
$\mathop{\mathrm{dim}}\nolimits X$ is even, and $\rho^{\prime}$ is the adjoint
representation, then the torsion function is locally constant; see [Söz, Wit].
Meanwhile, if $X$ is a knot complement in the 3-sphere with a tori boundary,
torsions from some adjoint representations are shown to be locally constant;
see [PY].

Furthermore, we observe that, if $\rho$ is the adjoint representation with
$G={\rm SU}(n)$ and $\mathop{\mathrm{dim}}\nolimits X=3$, the map (8) can be
regarded as a volume form of an open dence subset of $R_{G}(X)$ as follows.
Let ${\mathbb{F}}$ be ${\mathbb{R}}$, and $\psi$ be the killing form. From the
viewpoint of a stratification of $R_{G}(X)$ as a real algebraic variety, there
is an open dense set $R_{G}(X)^{\rm o}\subset R_{G}(X)$ which has a
$C^{\infty}$-manifold structure; see, e.g., [Kar]. As a classical result,
recall that, for $f\in R_{G}(X)^{\rm o}$, the first cohomology
$H^{1}_{\rho^{\prime}\circ f}(M;\mathfrak{g})$ can be identified with the
tangent space of $R_{G}(X)^{\rm o}$, as in [Sic, Kar]. Because $G$ is semi-
simple, the killing form $\psi$ is nondegenerate; thus, the pairing (5) is
nondegenerate by Proposition 3.2. Thus, the correspondence

$H^{2}_{\rho^{\prime}\circ
f}(X;\mathfrak{g})\longrightarrow\mathop{\mathrm{Hom}}\nolimits(H^{1}_{\rho^{\prime}\circ
f}(X;\mathfrak{g}),{\mathbb{R}});\ \ w\longmapsto(v\mapsto\psi((v\smile
w)\cap[X]))$

is as isomorphism, and therefore, the second one $H^{2}_{\rho^{\prime}\circ
f}(X;\mathfrak{g})$ can be identified with the cotangent space of
$R_{G}(X)^{\rm o}$. Let $d$ be the dimension
$\mathop{\mathrm{dim}}\nolimits(H^{2}_{\rho^{\prime}\circ
f}(X;\mathfrak{g}))$. Thus, for a basis ${\mathbf{h}}_{2}$ of
$H^{2}_{\rho^{\prime}\circ f}(X;\mathfrak{g})$, the $d$-th exterior product
(i.e., the determinant bundle) of ${\mathbf{h}}_{2}$ is a $d$-form of the
manifold $R_{G}(X)^{\rm o}$. Let ${\mathbf{h}}_{2}^{\otimes}$ denote the
$d$-form, and $\mathbf{h}_{2}^{\rm dual}$ be the dual basis of
$\mathbf{h}_{2}$ by Proposition 3.2. From (7), we can easily show that
$|\tau_{\rho^{\prime}\circ f}^{0}(X,\\{\mathbf{h}_{2},\mathbf{h}_{2}^{\rm
dual}\\}|^{1/2}\mathbf{h}_{2}^{\otimes}$ in
$\bigwedge^{d}\mathop{\mathrm{dim}}\nolimits(H^{2}(X;\mathfrak{g}))$ is
independent of the choices of $\mathbf{h}_{2}$. In summary, the dual of the
correspondence $f\mapsto|\tau_{\rho^{\prime}\circ
f}^{0}(X,\\{\mathbf{h}_{2},\mathbf{h}_{2}^{\rm
dual}\\})|^{1/2}\mathbf{h}_{2}^{\otimes}$ yields a volume form of the variety
$R_{G}(X)^{\rm o}$. Thus, it is reasonable to define the volume of the
$SU(n)$-character variety as follows:

######  Definition 4.2.

Assume that $\mathop{\mathrm{dim}}\nolimits(X)=3$ and $G={\rm SU}(n)$. We
define the volume of $R_{G}(X)^{\rm o}$ to be the integral of
$|\tau_{\rho^{\prime}\circ f}^{0}(X,\\{\mathbf{h}_{2},\mathbf{h}_{2}^{\rm
dual}\\})|^{1/2}\mathbf{h}_{2}^{\otimes}$ on $R_{G}(X)^{\rm o}$.

In §8.3, we provide examples of computing the volume, where $X$ is a torus
bundle.

Finally, we discuss a comparison with the preprint [MP]. Here, starting from
the same assumption in Definition 4.2, the authors define a volume of the
variety $R_{G}(X)^{\rm o}$ using another approach (see Theorems 3.5 and 3.6 in
[MP]) and examine some relations with [Wit]. However, their definition
requires a certain condition; but Definition 4.2 does not. It is seemingly
interesting to check if their volume and our volume in the condition are
equal.

##  5 Some computations of the quotient groups
${\mathbb{F}}^{\times}/N({\mathbb{F}})$

The torsion (6) lies in the quotient group
${\mathbb{F}}^{\times}/N({\mathbb{F}})$ in the case
$\mathrm{Im}(\rho)\subset\mathrm{SL}_{n}({\mathbb{F}})$, and therefore, we
provide some computations of the groups of some fields ${\mathbb{F}}$.

######  Example 5.1.

  1. (i)

If equation $x\bar{x}=1$ implies $x=1\in{\mathbb{F}}$, then
${\mathbb{F}}^{\times}/N({\mathbb{F}})$ is trivial. For example, if
$\bar{x}=x$ for any $x\in X$ and ${\mathbb{F}}$ is quadratically closed, then
${\mathbb{F}}^{\times}/N({\mathbb{F}})=0.$

  2. (ii)

Let $q$ be a prime power, and let ${\mathbb{F}}$ be the finite field of order
$q$. Since ${\mathbb{F}}^{\times}$ is a cyclic group,
${\mathbb{F}}^{\times}/N({\mathbb{F}})\cong{\mathbb{Z}}/2$ if $q$ is odd, and
${\mathbb{F}}^{\times}/N({\mathbb{F}})\cong 0$ if $q$ is even.

  3. (iii)

Let ${\mathbb{F}}$ be ${\mathbb{C}}$, and the involution be the complex
conjugacy. Then, the map ${\mathbb{C}}^{\times}\rightarrow S^{1}$ that sends
$x$ to $x/|x|$ induces the isomorphism
${\mathbb{C}}^{\times}/N({\mathbb{C}})\cong S^{1}=U(1)$.

Next, we focus on the case where ${\mathbb{F}}$ is the fraction field of a
unique factorization domain $A$ satisfying $\overline{A}\subset A$. We choose
a complete representative set of all prime elements of $A$ and denote the set
by $\mathfrak{P}_{A}$. Notice $p\in A$ is prime if and only if $\bar{p}$ is
prime. Without loss of generality, we can suppose that every
$p\in\mathfrak{P}_{A}$ satisfies either $\bar{p}\in\mathfrak{P}_{A}$ or
$\bar{p}=a_{p}p$ for some $a_{p}\in A^{\times}$. Let
$\mathfrak{P}_{A}^{{\mathbb{Z}}/2}$ be the subset $\\{p\in\mathfrak{P}_{A}\mid
p^{-1}\bar{p}\in A^{\times}\\}$. We can choose a subset
$\mathfrak{cP}_{A}\subset\mathfrak{P}_{A}$ such that
$\mathfrak{P}_{A}=\mathfrak{P}_{A}^{{\mathbb{Z}}/2}\sqcup\mathfrak{cP}_{A}\sqcup\overline{\mathfrak{cP}_{A}}.$
Then, we have the isomorphism

$\bigoplus_{q\in\mathfrak{P}_{A}^{{\mathbb{Z}}/2}}{\mathbb{Z}}/2\oplus\bigoplus_{p\in\mathfrak{cP}_{A}}{\mathbb{Z}}\cong{\mathbb{F}}^{\times}/\langle
N({\mathbb{F}}),A^{\times}\rangle,$ (9)

which sends $((a_{p})_{p},(b_{q})_{q})$ to
$\prod_{p}p^{a_{p}}\prod_{q}q^{b_{q}}$. Further, we also have a short exact
sequence

$0\rightarrow A^{\times}/\\{a\bar{a}\mid a\in
A^{\times}\\}\longrightarrow{\mathbb{F}}^{\times}/N({\mathbb{F}})\longrightarrow{\mathbb{F}}^{\times}/\langle
N({\mathbb{F}}),A^{\times}\rangle\rightarrow 0,$ (10)

where the first (resp. second) map is obtained by inclusion
$A\hookrightarrow{\mathbb{F}}$ (resp. projection). In summary, from (9) and
(10), we can compute the group ${\mathbb{F}}^{\times}/N({\mathbb{F}})$ as
follows:

######  Example 5.2.

Suppose that the involution is the identity. Then, $\mathfrak{cP}_{A}$ is
obviously empty, and $p=\bar{p}$ for any $p\in\mathfrak{P}_{A}$; thus, the
sequence (10) splits. Hence,

${\mathbb{F}}^{\times}/N({\mathbb{F}})={\mathbb{F}}^{\times}/({\mathbb{F}}^{\times})^{2}\cong(A^{\times}/(A^{\times})^{2})\oplus\bigoplus_{p\in\mathfrak{P}_{A}}{\mathbb{Z}}/2.$

For example, if ${\mathbb{F}}={\mathbb{Q}}$ and $A={\mathbb{Z}}$, then
${\mathbb{Q}}^{\times}/N({\mathbb{Q}})\cong\bigoplus_{p\in\mathrm{Spec}({\mathbb{Z}})}{\mathbb{Z}}/2$.
Furthermore, if ${\mathbb{F}}={\mathbb{R}}$, then
${\mathbb{F}}^{\times}/N({\mathbb{F}})\cong{\mathbb{Z}}/2$.

######  Example 5.3.

As mentioned in §4, let ${\mathbb{F}}$ be the fractional field
${\mathbb{Q}}(t_{1},\dots,t_{\ell})$ over ${\mathbb{Q}}$. Here, the involution
of $f(t_{1},\dots,t_{\ell})\in{\mathbb{F}}$ is defined by
$f(t_{1}^{-1},\dots,t_{\ell}^{-1})$. ${\mathbb{F}}$ is the fraction field of
$A={\mathbb{Q}}[t^{\pm 1}_{1},\dots,t_{\ell}^{\pm 1}]$ as a unique
factorization domain. We say $f(t_{1},\dots,t_{\ell})\in{\mathbb{F}}$ to be
reciprocal if $\bar{f}=rt_{1}^{n_{1}}\cdots t_{\ell}^{n_{\ell}}f$ for some
$r\in{\mathbb{Q}}^{\times},n_{i}\in{\mathbb{Z}}$. Then, the subset
$\mathfrak{P}_{A}^{{\mathbb{Z}}/2}$ is the set consisting of reciprocal
irreducible polynomials in ${\mathbb{Q}}[t^{\pm 1}_{1},\dots,t_{\ell}^{\pm}]$,
and any $p\in\mathfrak{cP}_{A}$ is not reciprocal. The group
$A^{\times}/\\{a\bar{a}\mid a\in A^{\times}\\}$ is isomorphic to
${\mathbb{Q}}^{\times}/({\mathbb{Q}}^{\times})^{2}\oplus{\mathbb{Z}}^{\ell}$,
where ${\mathbb{Z}}^{\ell}$ is generated by $t_{1},\dots,t_{\ell}$; the
sequence (10) turns out to be

$0\rightarrow{\mathbb{Q}}^{\times}/({\mathbb{Q}}^{\times})^{2}\oplus{\mathbb{Z}}^{\ell}\rightarrow{\mathbb{Q}}(t_{1},\dots,t_{\ell})^{\times}/N({\mathbb{Q}}(t_{1},\dots,t_{\ell}))\rightarrow\bigoplus_{p\in\mathfrak{P}_{A}^{{\mathbb{Z}}/2}}{\mathbb{Z}}/2\oplus\bigoplus_{q\in\mathfrak{cP}_{A}}{\mathbb{Z}}\rightarrow
0.$ (11)

Here, $1\neq-1$ and $(g-\bar{g})^{2}=-(g-\bar{g})\overline{(g-\bar{g})}=-1$
hold in ${\mathbb{F}}^{\times}/N({\mathbb{F}})$ for any $g\in A$ such that
$g\neq\bar{g}$; Thus, ${\mathbb{F}}^{\times}/N({\mathbb{F}})$ contain many
elements of order 4, unless $\ell=1$. The exact sequence does not split.

We proceed with the computation for the simplest case of $\ell=1$. Since
${\mathbb{Q}}[t]$ is a Euclidean domain, any polynomial
$f\in{\mathbb{Q}}[t^{\pm 1}]$ ensures some
$n,m,u,s\in{\mathbb{Z}},r\in{\mathbb{Q}},a_{i}\in\mathfrak{P}_{A}^{{\mathbb{Z}}/2}$
and $b_{j}\in\mathfrak{P}_{A}$ such that $f=r(1-t)^{u}(1+t)^{s}a_{1}\cdots
a_{n}b_{1}\cdots b_{m}$ and $\overline{a_{i}}=a_{i}$. Further,
$(1-t)^{2}=-t(1-t)(1-\bar{t})$ and $(1+t)^{2}=t(1+t)(1+\bar{t})$. Thus, the
subgroup of ${\mathbb{Q}}(t)^{\times}/N({\mathbb{Q}}(t))$ generated by
$-1,t\pm 1,t$ is isomorphic to ${\mathbb{Z}}\oplus{\mathbb{Z}}/4$, where
${\mathbb{Z}}$ is generated by $t+1$ and ${\mathbb{Z}}/4$ is obtained by
$(t-1)(1+t^{-1})$. Let $\mathfrak{sP}_{A}$ be the set of symmetric irreducible
polynomials in ${\mathbb{Q}}[t^{\pm 1}]$. In conclusion, we obtain from (11)
an isomorphism

${\mathbb{Q}}(t)^{\times}/N({\mathbb{Q}}(t))\cong{\mathbb{Z}}\oplus{\mathbb{Z}}/4\oplus\bigoplus_{p\in\mathrm{Spec}({\mathbb{Z}})\setminus\\{0\\}}{\mathbb{Z}}/2\oplus\bigoplus_{p\in\mathfrak{sP}_{A}\setminus\\{t\pm
1\\}}{\mathbb{Z}}/2\oplus\bigoplus_{q\in\mathfrak{cP}_{A}}{\mathbb{Z}}.$

######  Example 5.4.

Suppose that ${\mathbb{F}}\subset{\mathbb{C}}$ is a number field and the
involution is the complex conjugacy satisfying
$\overline{{\mathbb{F}}}\subset{\mathbb{F}}$. Let
$\mathcal{O}_{{\mathbb{F}}}\subset{\mathbb{F}}$ be the integral closure of
${\mathbb{Z}}$. Dirichlet’s unit theorem indicates that the first term in (10)
is computed as

$\mathcal{O}_{{\mathbb{F}}}^{\times}/\\{a\bar{a}\mid
a\in\mathcal{O}_{{\mathbb{F}}}^{\times}\\}\cong({\mathbb{Z}}/2{\mathbb{Z}})^{r_{1}-1}\oplus{\mathbb{Z}}^{r_{2}}\oplus(\\{a\in{\mathbb{F}}\mid
a\bar{a}=1\\}/({\mathbb{Z}}/2{\mathbb{Z}})),$

where $r_{1}$ represents the number of real embeddings and $r_{2}$ represents
the number of conjugate pairs of complex embeddings of ${\mathbb{F}}$. The
latter quotient by ${\mathbb{Z}}/2{\mathbb{Z}}$ is defined by the involution.
Recall also that the ideal class group $C(\mathcal{O}_{{\mathbb{F}}})$ is
trivial if and only if $\mathcal{O}_{{\mathbb{F}}}$ is a UFD. Thus, if
$C(\mathcal{O}_{{\mathbb{F}}})$ is trivial, then we can compute the group
${\mathbb{F}}^{\times}/N({\mathbb{F}})$ by (9) and (10).

However, if $C(\mathcal{O}_{{\mathbb{F}}})\neq 1$, it is difficult to obtain a
similar isomorphism to (9) because of the following reasons. It follows from
Hasse’s local-global principle that the computation of
${\mathbb{F}}^{\times}/N({\mathbb{F}})$ is derived from those of
${\mathbb{F}}_{\mathfrak{p}}^{\times}/(N({\mathbb{F}}_{\mathfrak{p}}))$, where
$\mathfrak{p}\in\mathrm{Spec}(\mathcal{O}_{{\mathbb{F}}})$ and
${\mathbb{F}}_{\mathfrak{p}}$ is the local field. However, even if the residue
field of $\mathfrak{p}$ is of characteristic two, the computation of
${\mathbb{F}}_{\mathfrak{p}}^{\times}/(N({\mathbb{F}}_{\mathfrak{p}}))$ is
difficult. Furthermore, if ${\mathbb{Q}}\subset{\mathbb{F}}$ is not an abelian
extension, it is also difficult to determine all prime ideals of
$\mathcal{O}_{{\mathbb{F}}}$ even using the class field theory.

##  6 Some topological invariants from refined torsions

As corollaries of Theorem 3.1, we introduce some topological invariants in
terms of refined torsions. For example, we show that the theorem produces
extended definitions of abelian Reidemeister torsions and (multivariable)
Alexander polynomials of links; see Sections 6.3 and 6.4.

###  6.1 Torsions from the trivial coefficients over a field ${\mathbb{F}}$

We start by examining the case where $\rho$ with $n=1$ is trivial and $\psi$
represents the canonical multiplication of ${\mathbb{F}}$. Then,
$H^{j}(X;{\mathbb{F}})$ represents the ordinal cohomology of $X$; the ordinal
cup product in (5) is widely known to be nondegenerate.

######  Definition 6.1.

Let $X$ be an oriented closed manifold of dimension $2m+1$. We define the
torsion of $X$ to be the refined torsion (6) in
${\mathbb{F}}^{\times}/({\mathbb{F}}^{\times})^{2}$.

######  Example 6.2.

Suppose that $X$ admits a CW-complex structure that all boundary maps of the
cellular complex is zero. Then, $\mathbf{h}_{j}=\mathbf{c}_{j}$, and
$\mathbf{b}_{j}=\emptyset$ by definitions. Thus, to compute the refined
torsion, we may only find a dual basis $\mathbf{h}_{j}^{\rm dual}$ by
computing the cup product and $[\mathbf{h}_{j}^{\rm
dual}/\mathbf{c}_{2m+1-j}]\in{\mathbb{F}}^{\times}$. In particular, the
refined torsion is a weaker invariant than the cup product structure of $X$ in
coefficients ${\mathbb{F}}$.

However, we provide two examples. Let $X$ be the torus $(S^{1})^{2m+1}$.
Consider the CW-structure of $S^{1}$ consisting of a $1$-cell and a $0$-cell
and the product CW-structure of $X$. Then, by an easy computation of the cross
product, we can check that $\mathbf{h}_{j}^{\rm dual}=\mathbf{c}_{2m+1-j}$.
Hence, the torsion is trivially 1.

Let ${\mathbb{F}}$ be the finite field ${\mathbb{F}}_{p}$ and $X$ be the
three-dimensional lens space $L(p,q)$, where $p$ is an odd prime and
$q\in{\mathbb{Z}}$. Then, by the genus-one Heegaard decomposition, we have a
CW-structure of $X$ such that the $k$-cell is single with $0\leq k\leq 3$; the
boundary map is zero. Then, by the resulting computation of the cup product
(see, e.g., [Nos, Example 5.1]), $\mathbf{h}_{1}^{\rm dual}=q\mathbf{c}_{2}$
and $\mathbf{h}_{0}^{\rm dual}=\mathbf{c}_{3}$. In conclusion, the refined
torsion is
$q\in{\mathbb{F}}_{p}^{\times}/({\mathbb{F}}_{p}^{\times})^{2}\cong{\mathbb{Z}}/2.$
The torsions distinguish between $L(7,1)$ and $L(7,2)$.

###  6.2 Torsions from the trivial coefficients over ${\mathbb{Q}}$

We focus on the rational case ${\mathbb{F}}={\mathbb{Q}}$ and suggest a
modification of Definition 6.1. For a finitely generated abelian group $A$,
$\mathrm{Free}(A)$ denotes the free submodule of $A$. The inclusion
$\iota:{\mathbb{Z}}\hookrightarrow{\mathbb{Q}}$ induces an isomorphism
$\iota_{*}:H^{*}(X;{\mathbb{Z}})\otimes{\mathbb{Q}}\cong
H^{*}(X;{\mathbb{Q}})$, and the cup product
$\smile:\mathrm{Free}H^{j}(X;{\mathbb{Z}})\times\mathrm{Free}H^{2m+1-j}(X;{\mathbb{Z}})\rightarrow{\mathbb{Z}}$
is unimodular, i.e., the determinant of $\smile$ is $+1$.

######  Proposition 6.3.

Choose a basis ${\mathbf{h}}_{j}$ of $\mathrm{Free}H^{j}(X;{\mathbb{Z}})$ for
$j\leq m$. Let ${\mathbf{h}}_{j}^{\rm dual}\subset H^{2m+1-j}(X;{\mathbb{Z}})$
be the dual basis of ${\mathbf{h}}_{j}$. The refined torsion
$\tau^{0}_{\rho}(X,{\mathbf{h}})$ in (6) valued in ${\mathbb{Q}}$ is a
topological invariant of $X$.

Since the proof is almost the same as the proof of Theorem 3.1, we have not
presented it here. However, if $X$ is a rational homology sphere or if
$\mathop{\mathrm{dim}}\nolimits X=3$, then the torsion is not entirely
interesting as follows:

######  Proposition 6.4.

  1. (i)

If $H^{*}(X;{\mathbb{Q}})\cong H^{*}(S^{2m+1};{\mathbb{Q}})$, then the torsion
$\tau^{0}_{\rho}(X,{\mathbf{h}})$ is equal to
$\prod_{i=1}^{2m}|H^{i}(X;{\mathbb{Z}})|^{(-1)^{i}}\in{\mathbb{Q}}$.

  2. (ii)

If $m=1$, then the torsion $\tau^{0}_{\rho}(X,{\mathbf{h}})$ equals
$\varepsilon_{X}|H^{1}(X;{\mathbb{Q}}/{\mathbb{Z}})|^{-1}\in{\mathbb{Q}}$ for
some $\varepsilon_{X}\in\\{\pm 1\\}$.

###### Proof.

By Morse theory, we can choose a CW complex on $X$ such that the $0$-th and
$(2m+1)$-th cells are single. Then, the boundary maps $\partial_{0}$ and
$\partial_{2m}$ in the cellular complex $C^{*}(X;{\mathbb{Z}})$ are zero.
Consider the subcomplex $E^{*}$ defined by $C^{*}(X;{\mathbb{Z}})$ if
$1\leq*\leq 2m$ and by zero otherwise. If $H^{*}(X;{\mathbb{Q}})\cong
H^{*}(S^{2m+1};{\mathbb{Q}})$, then $E^{*}\otimes{\mathbb{Q}}$ is acyclic, and
$\tau^{0}_{\rho}(X,{\mathbf{h}})$ is equal to the torsion of
$E^{*}\otimes{\mathbb{Q}}$; Theorem 1.4 in [Tur] immediately implies that the
torsion of $E^{*}\otimes{\mathbb{Q}}$ is equal to
$\prod_{i=1}^{2m}|H^{i}(X;{\mathbb{Z}})|^{(-1)^{i}}\in{\mathbb{Q}}$. Hence, we
complete the proof of (i).

Next, consider the case $m=1$, i.e., $\mathop{\mathrm{dim}}\nolimits X=3.$
Then, the subcomplex is presented as $0\rightarrow
C^{1}(X;{\mathbb{Z}})\stackrel{{\scriptstyle\partial_{1}}}{{\longrightarrow}}C^{2}(X;{\mathbb{Z}})\rightarrow
0.$ Suppose that the elementary divisors of $\partial_{1}$ are
$e_{1},\dots,e_{k},0,\dots,0$ for some $e_{i}\in\mathbb{N}$. Note that
$|H^{1}(X;{\mathbb{Q}}/{\mathbb{Z}})|=|H_{2}(X;{\mathbb{Z}})/\mathrm{Free}(H_{2}(X;{\mathbb{Z}}))|=e_{1}e_{2}\cdots
e_{k}.$ The elementary divisor theory implies that $\partial_{1}$ is the
diagonal matrix $e_{1}\oplus\cdots\oplus e_{k}\oplus 0\oplus\cdots\oplus 0$ up
to the choice of the base of $C^{2}(X;{\mathbb{Z}})$. Since $Z^{1}=H^{1}$, we
can verify that the torsion of the diagonal matrix is equal to
$\varepsilon_{X}(e_{1}\cdots e_{k})^{-1}$. In summary, the required torsion is
equal to $\varepsilon_{X}|H^{1}(X;{\mathbb{Q}}/{\mathbb{Z}})|^{-1}$ as claimed
in (ii). ∎

###  6.3 Nonacyclic abelian Reidemeister torsions

The torsions in Theorem 3.1 in abelian coefficients yield an extended
definition of the abelian Reidemeister torsions including nonacyclic cases.

First, we review abelian Reidemeister torsions from [Tur, Chapter II]. We
choose the decomposition
$\mu:H_{1}(X;{\mathbb{Z}})\cong\mathrm{Free}(H_{1}(X))\oplus\mathrm{Tor}(H_{1}(X))$
and a generating set $\\{t_{1},\dots,t_{\ell}\\}$ of
$\mathrm{Free}(H_{1}(X))$. Let $\rho_{\rm ab}$ be the composite of the
abelianization of $\pi_{1}(X)$ with $\mu$ and the map
$\mathrm{Free}(H_{1}(X))\rightarrow{\rm GL}_{1}({\mathbb{F}})$ that sends
$t_{i}$ to $t_{i}$ itself. Let ${\mathbb{F}}$ be the field
${\mathbb{Q}}(t_{1},\dots,t_{\ell})$ as in Example 5.3, which is a local
coefficient via $\rho_{\rm ab}$. If the complex $C^{*}_{\rho}(X;{\mathbb{F}})$
is acyclic, then the torsion of $C^{*}_{\rho}(X;{\mathbb{F}})$ is called the
maximal abelian torsion, and it is well-studied together with its relationship
to the Alexander polynomial and Milnor torsions; see [Mil, Tur].

On the other hand, concerning the torsions parts, it is known (see [Tur,
Theorem 12.5]) that the group ring ${\mathbb{Q}}[\mathrm{Tor}(H_{1}(X))]$ over
${\mathbb{Q}}$ is isomorphic to a direct sum of some cyclotomic fields
${\mathbb{F}}_{1}\oplus\cdots\oplus{\mathbb{F}}_{m}$, where
${\mathbb{F}}_{i}={\mathbb{Q}}(\mu_{i})\subset{\mathbb{C}}$ for some root of
unity $\mu_{i}$. Let
$p_{i}:{\mathbb{Q}}[\mathrm{Tor}(H_{1}(X))]\rightarrow{\mathbb{F}}_{i}$ be the
projection. Then, the composite of the abelianization of $\pi_{1}(X)$ and
$p_{i}$ gives the representation
$\rho_{i}:\pi_{1}(X)\rightarrow{\mathbb{F}}_{i}^{\times}={\rm
GL}_{1}({\mathbb{F}}_{i})$.

We explain the above purpose in detail. For
$f=f(t_{1},\dots,t_{\ell})\in{\mathbb{F}}$, we define the involution $\bar{f}$
as $f(t_{1}^{-1},\dots,t_{\ell}^{-1})$, and similarly, the involution on
${\mathbb{Q}}[\mathrm{Tor}(H_{1}(X))]$ by the minus multiplication of
$H_{1}(X)$. Define
$\psi:{\mathbb{F}}\times{\mathbb{F}}\rightarrow{\mathbb{F}}$ and
$\psi:{\mathbb{F}}_{i}\times{\mathbb{F}}_{i}\rightarrow{\mathbb{F}}_{i}$ by
$(a,b)\mapsto a\bar{b}$, which satisfy the nondegeneracy in Proposition 3.2.
Therefore, as a result of Theorem 3.1, even if $C^{*}_{\rho}(X;{\mathbb{F}})$
is not acyclic, we can define the torsion in

$\tau_{\rho_{\rm ab}}(X)\in{\mathbb{F}}/\langle
N({\mathbb{F}}),t_{1},\dots,t_{\ell}\rangle$ (12)

similar to (6). For the torsion subgroup of $H_{1}(X;{\mathbb{Z}})$, we can
define the torsion to be the tuple

$\bigl{(}\tau_{\rho_{1}}(X,\mathbf{h}),\dots,\tau_{\rho_{m}}(X,\mathbf{h})\bigr{)}\in{\mathbb{F}}_{1}^{\times}/\langle\mu_{1},N({\mathbb{F}}_{1})\rangle\times\cdots\times{\mathbb{F}}_{m}^{\times}/\langle\mu_{m},N({\mathbb{F}}_{m})\rangle,$

as in [Tur, Section II. 13]. In conclusion, these torsions in nonacyclic cases
can be interpreted as an extension of abelian Reidemeister torsions.

###  6.4 Extension of the Alexander polynomial of links

We define torsions of any link whose Alexander polynomial is zero as a
generalization of the Alexander polynomials of links.

For this, let us review the Alexander polynomial; see, e.g., [Tur, §II. 15].
Let $L\subset S^{3}$ be a link of $\ell$-components. We fix an abelianization
$\mu:\pi_{1}(S^{3}\setminus L)\rightarrow{\mathbb{Z}}^{\ell}$ and the
generators $t_{1},\dots,t_{\ell}$ of ${\mathbb{Z}}^{\ell}$. Let ${\mathbb{F}}$
be the fractional field ${\mathbb{Q}}(t_{1},\dots,t_{\ell})$. Since $\mu$ is
regarded as a representation $\pi_{1}(S^{3}\setminus L)\rightarrow{\rm
GL}_{1}({\mathbb{Z}}[t_{1}^{\pm 1},\dots,t_{\ell}^{\pm 1}])$ that sends $g$ to
$\mu(g)$, we have the homology $H_{*}(S^{3}\setminus L;{\mathbb{Z}}[t_{1}^{\pm
1},\dots,t_{\ell}^{\pm 1}])$ of local systems. Then, the (multivariable)
Alexander polynomial in ${\mathbb{Z}}[t_{1}^{\pm 1},\dots,t_{\ell}^{\pm
1}]/\\{\pm t_{1}^{k_{1}}\cdots
t_{\ell}^{k_{\ell}}\mid(k_{1},\dots,k_{\ell})\in{\mathbb{Z}}^{\ell}\\}$ is
defined to be the order of $H_{1}(S^{3}\setminus L;{\mathbb{Z}}[t_{1}^{\pm
1},\dots,t_{\ell}^{\pm 1}])$, that is, the minimal polynomial that annihilates
$H_{1}(S^{3}\setminus L;{\mathbb{Z}}[t_{1}^{\pm 1},\dots,t_{\ell}^{\pm 1}])$.
The polynomial is usually denoted by $\Delta_{L}(t_{1},\dots,t_{\ell})$. For
$t=t_{1}=\cdots=t_{\ell}$, the one variable polynomial
$\Delta_{L}(t,\dots,t)\in{\mathbb{Q}}(t)/\\{t^{k}\\}_{k\in{\mathbb{Z}}}$ is
also called the Alexander polynomial. The polynomial
$\Delta_{L}(t_{1},\dots,t_{\ell})$ is not zero if and only if the homology
$H_{1}(S^{3}\setminus L;{\mathbb{F}})$ vanishes, and $\Delta_{L}(t,\dots,t)$
is not zero if and only if the homology $H_{1}(S^{3}\setminus
L;{\mathbb{Q}}(t))$ vanishes. If $\ell=1$, that is, $L$ is any knot, then
$\Delta_{L}(t)\neq 0$ is well known.

We also review (13) below, which is shown by Turaev. Let $M_{L}$ be the closed
3-manifold obtained by the zero-surgery on $L$. Then, the abelianization $\mu$
induces an abelianization $\pi_{1}(M_{L})\rightarrow{\mathbb{Z}}^{\ell}$,
which is considered as representations $\mu_{\rm
ab}:\pi_{1}(M_{L})\rightarrow{\rm GL}_{1}({\mathbb{F}})$ or $\mu_{\rm
total}:\pi_{1}(M_{L})\rightarrow{\rm GL}_{1}({\mathbb{Q}}(t))$. Thus,
${\mathbb{F}}$ and ${\mathbb{Q}}(t)$ can be regarded as local coefficients of
$M_{L}.$ If the Alexander polynomial $\Delta_{L}(t_{1},\dots,t_{\ell})$ is not
zero (see [Tur, Corollary 15.2 and Theorem 17.2]), then

$\tau_{\mu}^{0}(M_{L})=\Delta_{L}(t_{1},\dots,t_{\ell})\prod_{j:1\leq
j\leq\ell}(t_{j}-1)\in{\mathbb{F}}^{\times}/\\{rt_{1}^{k_{1}}\cdots
t_{\ell}^{k_{\ell}}\mid(k_{1},\dots,k_{\ell})\in{\mathbb{Z}}^{\ell},r\in{\mathbb{Q}}^{\times}\\},$
(13)

and the same equality with $t=t_{1}=\cdots=t_{\ell}$ holds if
$\Delta_{L}(t,\dots,t)\neq 0$.

It seems sensible to extend the Alexander polynomial from the viewpoint of
refined torsions.

######  Definition 6.5.

Suppose $\mu>1$. Let ${\mathbb{F}}$ be ${\mathbb{Q}}(t_{1},\dots,t_{\ell})$ as
above. If $\Delta_{L}(t_{1},\dots,t_{\ell})$ is zero, we define an (extended)
Alexander polynomial of $L$ to be the torsion

$\tau^{0}_{\mu_{\rm ab}}(M_{L})\in{\mathbb{F}}^{\times}/\langle
N({\mathbb{F}}),\\{t_{1}^{k_{1}}\cdots
t_{\ell}^{k_{\ell}}\mid(k_{1},\dots,k_{\ell})\in{\mathbb{Z}}^{\ell}\\}\rangle,$
(14)

where $\tau^{0}_{\mu_{\rm ab}}$ is defined in (12). Furthermore, if
$\Delta_{L}(t,\dots,t)$ is zero, we define an (extended) Alexander polynomial
to be the torsion
$\tau^{0}_{\mu}(M_{L})\in{\mathbb{Q}}(t)^{\times}/\\{f(t)f(t^{-1})t^{n}\mid
f\in{\mathbb{Q}}(t),n\in{\mathbb{Z}}\\}.$

Recall that the quotient group in (14) is computed in Example 5.3. However,
establishing a procedure for computing the torsions of links with Alexander
polynomial zero remains a future problem.

##  7 Further refinements of the torsions with $b_{1}\geq 1$

In the preceding sections, we defined the torsions modulo the subgroup
$N({\mathbb{F}})$. This section restricts to 3-manifolds and suggests further
refinements of the torsion without using $N({\mathbb{F}})$; see Propositions
7.2 and 7.3. Throughout this section, we assume a closed 3-manifold $X$ and an
epimorphism $\alpha:\pi_{1}(X)\rightarrow{\mathbb{Z}}$. The first Betti number
of $X$ is larger than zero. By Poincaré duality $H_{2}(X;{\mathbb{Z}})\cong
H^{1}(X;{\mathbb{Z}})\cong\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(X),{\mathbb{Z}})$,
we can choose a homology 2-class $\Sigma_{\alpha}\in H_{2}(X;{\mathbb{Z}})$ as
a dual of $\alpha$.

######  Definition 7.1.

As in Theorem 3.1, we consider a representation
$\rho:\pi_{1}(X)\rightarrow{\rm GL}_{n}({\mathbb{F}})$ and a non-degenerate
(anti-)hermitian $\rho$-invariant bilinear form
$\psi:{\mathbb{F}}^{n}\times{\mathbb{F}}^{n}\rightarrow{\mathbb{F}}$. Then,
the triple $(\rho,\psi,\Sigma)$ is said to be admissible, if the following
pairing and (5) are nondegenerate:

$H^{1}_{\rho}(X;{\mathbb{F}}^{n})\otimes
H^{1}_{\rho}(X;{\mathbb{F}}^{n})\stackrel{{\scriptstyle\smile}}{{\longrightarrow}}H^{2}_{\rho}(X;{\mathbb{F}}^{n}\otimes{\mathbb{F}}^{n})\xrightarrow{\
\cap[\Sigma_{\alpha}]\
}H^{0}_{\rho}(X;{\mathbb{F}}^{n}\otimes{\mathbb{F}}^{n})\stackrel{{\scriptstyle\psi_{*}}}{{\longrightarrow}}{\mathbb{F}}.$
(15)

######  Proposition 7.2.

Suppose that the triple $(\rho,\psi,\Sigma)$ is admissible, and the zeroth
cohomology $H^{0}(X;{\mathbb{F}}^{n})$ vanishes. By the nondegeneracy, we
choose a normalized orthonormal basis ${\mathbf{h}}_{1}$ of
$H^{1}_{\rho}(X;{\mathbb{F}}^{n})$ with respect to (15), and the dual basis
${\mathbf{h}}_{1}^{\rm dual}$ of $H^{2}_{\rho}(X;{\mathbb{F}}^{n})$. Then, the
refined torsion

$\tau^{0}_{\rho}(X,\\{{\mathbf{h}}_{1},\mathbf{h}_{1}^{\rm
dual}\\})\in{\mathbb{F}}^{\times}/\pm\mathop{\mathrm{det}}\nolimits(\pi_{1}(X))$
(16)

modulo $\pm\mathop{\mathrm{det}}\nolimits(\pi_{1}(X))$ is independent of the
choice of the basis ${\mathbf{h}}_{1}$.

###### Proof.

Consider another orthonormal basis ${\mathbf{h}}_{1}^{\prime}$. Then, there is
an element $T$ of the unitary group
$U(\mathrm{dim}H^{1}(X;{\mathbb{F}}^{n});{\mathbb{F}})$ such that
$T{\mathbf{h}}_{1}={\mathbf{h}}_{1}^{\prime}$.
$[{\mathbf{h}}_{1}/{\mathbf{h}}_{1}^{\prime}]=\mathrm{det}(T)\in
N({\mathbb{F}})$ and $\mathrm{det}(T)\overline{\mathrm{det}(T)}=1.$ Thus, as
seen in the proof of Theorem 3.1,
$\mathrm{det}(T)=[{\mathbf{h}}_{1}/{\mathbf{h}}_{1}^{\prime}]=\overline{[{\mathbf{h}}_{1}^{\rm
dual}/({\mathbf{h}}_{1}^{\prime})^{\rm dual}]}^{-1}$. Hence,

$\tau^{0}_{\rho}(X,\\{{\mathbf{h}}_{1},\mathbf{h}_{1}^{\rm
dual}\\})=\tau^{0}_{\rho}(X,\\{{\mathbf{h}}_{1}^{\prime},(\mathbf{h}_{1}^{\rm
dual})^{\prime}\\})\frac{[{\mathbf{h}}_{1}/{\mathbf{h}}_{1}^{\prime}]}{[{\mathbf{h}}_{1}^{\rm
dual}/({\mathbf{h}}_{1}^{\prime})^{\rm
dual}]}=\tau^{0}_{\rho}(X,\\{{\mathbf{h}}_{1}^{\prime},(\mathbf{h}_{1}^{\rm
dual})^{\prime}\\})$

holds in
${\mathbb{F}}^{\times}/\pm\mathop{\mathrm{det}}\nolimits(\pi_{1}(X))$. This
implies the required independence. ∎

In general, there is no clarity on whether the existence of such a 2-class
$\Sigma_{\alpha}$ is admissible; see §8.2 for examples of such existences. For
example, even if $X$ is a fibered 3-manifold, it does not always hold that the
canonical map $\alpha:\pi_{1}(X)\rightarrow{\mathbb{Z}}$ yields an admissible
pairing (15); see Proposition 8.9 for details. Alternatively, we suggest
another refinement.

######  Proposition 7.3.

Assume that the inclusion $\Sigma_{\alpha}\hookrightarrow X$ induces an
isomorphism $H^{2}(X;{\mathbb{F}}^{n})\cong
H^{2}(\Sigma_{\alpha};{\mathbb{F}}^{n})\cong{\mathbb{F}}$ generated by the
fundamental class $[\Sigma_{\alpha}]$, and that the pairing (5) is
nondegenerate. Then, the following torsion depends only on
$(\rho,\psi,\alpha)$:

$\tau^{0}_{\rho}(X,\\{[\Sigma_{\alpha}]^{\rm
dual},[\Sigma_{\alpha}]\\})\in{\mathbb{F}}^{\times}/\pm\mathop{\mathrm{det}}\nolimits(\pi_{1}(X)).$
(17)

Since the proof can be obtained easily, we omit the detailed proof. The
assumptions in the above propositions are strong in many cases; however, we
later see that the refinement is of use for the adjoint torsions of the torus
bundles; see §8.3.

##  8 Some computations of the refined torsions of 3-manifolds

In general, computation of the refined torsions is not easy. In this section,
we focus on closed 3-manifolds and develop a three-dimensional procedure for
computing the refined torsions. That is, we fix $m=1$ and
$\mathop{\mathrm{dim}}\nolimits X=3.$

Section 8.1 reviews Heegaard splittings and cup products in terms of
identities, and Sections 8.2 and 8.3 compute some refined torsions. Throughout
this section, for integers $0<j\leq n$, let $f^{(n)}_{j}\in{\mathbb{F}}^{n}$
be the vector in which the $j$-th entry is $1$ and the other entries are
zeros; further, let $E$ represent an identity matrix.

###  8.1 Heegaard splittings and the presentation of the fundamental group

Let $X$ be an oriented closed 3-manifold, and let
$\rho:\pi_{1}(X)\rightarrow{\rm GL}_{n}({\mathbb{F}})$. Following [Tro], we
establish a procedure to compute cup products.

We choose a genus-$g$ Heegaard decomposition of $X.$ Then, $X$ consists of a
single 0-handle, $g$ pieces of 1-, 2-handles, and a single 3-handle. Thus,
using the von Kampen theorem, we have a group presentation $\langle
x_{1},\dots,x_{g}\mid r_{1},\dots,r_{g}\rangle$ of $\pi_{1}(X)$. Moreover, the
(augmented) cellular complex of the universal covering space of $X$ can be
described as

$C_{*}(\widetilde{X};{\mathbb{Z}}):0\rightarrow{\mathbb{Z}}[\pi_{1}(X)]\stackrel{{\scriptstyle\partial_{3}}}{{\longrightarrow}}{\mathbb{Z}}[\pi_{1}(X)]^{g}\stackrel{{\scriptstyle\partial_{2}}}{{\longrightarrow}}{\mathbb{Z}}[\pi_{1}(X)]^{g}\stackrel{{\scriptstyle\partial_{1}}}{{\longrightarrow}}{\mathbb{Z}}[\pi_{1}(X)]\rightarrow{\mathbb{Z}}\rightarrow
0,$ (18)

where ${\mathbb{Z}}[\pi_{1}(X)]$ represents the group ring of $\pi_{1}(X)$. It
is known that (see, e.g., [Tro]) the boundary maps $\partial_{1}$ and
$\partial_{2}$ can be described as

$\partial_{2}=\bigl{\\{}[\frac{\partial r_{j}}{\partial
x_{i}}]\bigr{\\}}_{1\leq i,j\leq g}\in\mathrm{Mat}(g\times
g;{\mathbb{Z}}[\pi_{1}(X)]),$ (19)
$\partial_{1}=(1-x_{1},1-x_{2},1-x_{3},\dots,1-x_{g})^{\rm transpose}.$ (20)

Here, $\frac{\partial r_{j}}{\partial x_{i}}$ represents the Fox derivative of
$r_{j}$ with respect to $x_{i}$.

Next, we review a description of the third map $\partial_{3}$. Let $F$ be the
free group $\langle x_{1},\dots,x_{g}\ |\ \rangle$ and $P$ be the free group
$\langle\rho_{1},\ldots,\rho_{g}\ |\ \rangle$. We define the group
homomorphism $\Psi:P*F\rightarrow F$ by $\Psi(\rho_{j})=r_{j},\
\Psi(x_{i})=x_{i}.$ An element $s\in P*F$ is an identity if
$s\in\mathop{\mathrm{Ker}}\nolimits(\Psi)$ and $s$ can be written as
$\prod_{m=1}^{\ell}w_{m}\rho_{j_{m}}^{\epsilon_{m}}w_{m}^{-1}$ for some
$w_{m}\in F,\ \epsilon_{m}\in\\{\pm 1\\},$ and $\rho_{j_{m}}\in P$.

######  Theorem 8.1 ([S80]; see also [Wak, Section 4]).

For any closed 3-manifold $X$, there exists an identity $W_{X}$ such that
$\partial_{3}=\Bigl{(}\mu\bigl{(}[\psi(\frac{\partial
W_{X}}{\partial\rho_{1}})]\bigr{)},\dots,[\psi(\frac{\partial
W_{X}}{\partial\rho_{g}})]\bigr{)}\Bigr{)}$. Here, $\mu$ is the natural
surjection from $F$ to $\pi_{1}(M)$.

In terms of the identity $W_{X}$, we provide the formula to determine the
pairing (5). Let $c_{3}$ represent the unit $1$ in
$C_{3}(\widetilde{X};{\mathbb{Z}})={\mathbb{Z}}[\pi_{1}(X)]$. If the identity
$W_{X}$ forms
$\prod_{k=1}^{\ell}\omega_{k}\rho_{j_{k}}^{\epsilon_{k}}\omega_{k}^{-1}$, we
define

$D^{\sharp}(c_{3})=\sum_{k=1}^{\ell}\epsilon_{k}\bigl{(}\sum_{\ell=1}^{m}[\frac{\partial\omega_{k}}{\partial
x_{\ell}}]a_{\ell}\otimes\omega_{k}b_{j_{k}}\bigr{)}\in
C_{1}(\widetilde{X};{\mathbb{Z}})\otimes C_{2}(\widetilde{X};{\mathbb{Z}}).$
(21)

Then, for cochains $p\in C^{1}_{\rho}(X;{\mathbb{F}}^{n})$ and $q\in
C^{2}_{\rho}(X;{\mathbb{F}}^{n})$, we define a 3-cochain $p\smile q$ by

$p\smile q(uc_{3}):=(p\otimes
q)(uD^{\sharp}(c_{3}))\in{\mathbb{F}}^{n}\otimes_{{\mathbb{F}}}{\mathbb{F}}^{n}.$

Here, $u\in{\mathbb{Z}}[\pi_{1}(X)]$. For a $\rho$-invariant bilinear map
$\psi:{\mathbb{F}}^{n}\otimes{\mathbb{F}}^{n}\rightarrow{\mathbb{F}}$, the map

$\smile:C^{1}_{\rho}(X;{\mathbb{F}}^{n})\otimes
C^{2}_{\rho}(X;{\mathbb{F}}^{n})\longrightarrow C^{3}_{\rho}(X;{\mathbb{F}});\
\ (p,q)\longmapsto\psi(p\smile q),$

induces the bilinear map on cohomology. Then, as a result of [Tro, §2.4], the
bilinear map is equal to the paring (5) with $(m,i)=(1,1)$.

Next, we review a procedure for computing the cup product (15) from [Tro,
§2.4]. We define $\alpha:F\rightarrow C_{1}(\widetilde{X};{\mathbb{Z}})$ and
$\kappa:F\times F\rightarrow C_{1}(\widetilde{X};{\mathbb{Z}})\otimes
C_{1}(\widetilde{X};{\mathbb{Z}})$ by

$\alpha(w):=\begin{pmatrix}\mu(\dfrac{\partial w}{\partial
x_{1}})&\mu(\dfrac{\partial w}{\partial x_{2}})&\cdots&\mu(\dfrac{\partial
w}{\partial x_{g}})\end{pmatrix}^{\rm transpose},\hskip
14.22636pt\kappa(u,v):=\alpha(u)\otimes\mu(u)\alpha(v).$

Then, we check if $\kappa$ is a 2-cocycle of $F$. Since
$H^{2}(F;C_{1}(\widetilde{X};{\mathbb{Z}})\otimes
C_{1}(\widetilde{X};{\mathbb{Z}}))=0$ by the freeness of $F$, we have uniquely
a map $\Upsilon:F\to C_{1}(\widetilde{X};{\mathbb{Z}})\otimes
C_{1}(\widetilde{X};{\mathbb{Z}})$ defined by the rules

$\Upsilon(uv)=\Upsilon(u)+\mu(u)\Upsilon(v)+\kappa(u,v),\hskip
14.22636pt\Upsilon(1)=0,\hskip 14.22636pt\Upsilon(x_{i})=0,$

where $u,v\in F$ and $i\in\\{1,\ldots,g\\}$. For any 1-cocycles
$f,f^{\prime}\in C^{1}_{\rho}(X;{\mathbb{F}}^{n})$ and 2-cycle
$c=(c_{1},c_{2},\ldots c_{g})\in{\mathbb{Z}}\otimes
C_{2}(\widetilde{X};{\mathbb{Z}})$,

$\psi_{*}((f\smile f^{\prime})\cap c)=\sum_{i=1}^{g}\psi\circ(f\otimes
f^{\prime})(c_{i}\Upsilon(r_{i}))\in{\mathbb{F}}.$ (22)

Consequently, when we can describe the 2-chain $c\in{\mathbb{Z}}\otimes
C_{2}(\widetilde{X};{\mathbb{Z}})$ concretely, we can compute the
(anti)-Hermitian form (15).

###  8.2 Computation 1; some Seifert 3-manifolds (uncompleted)

In this section, for three integers $(n,m,\ell)\in\mathbb{N}^{3}$ with
$\ell\geq m\geq n\geq 2$, we focus on the Seifert 3-manifold $X$ of the type
$M(0;(1,0);(\ell,-1),(m,-1),(n,1))$; see, e.g., [Kit] for the definition.
Then, there is a genus-two Heegaard splitting of $X$, which yields the group
presentation of $\pi_{1}(X)$ as $\langle g,h\ |\ r,s\rangle$ where
$r=(gh)^{n}h^{-m},s=(hg)^{n}g^{-\ell}$. Furthermore, the identity $W_{X}$ in
Theorem 8.1 is given by
$W_{X}=\rho_{r}h\rho_{r}^{-1}h^{-1}\rho_{s}g\rho_{s}^{-1}g^{-1}$ (see [S86,
p.127] and [Wak, Section 7]).

We now demonstrate all ${\rm SL}_{2}({\mathbb{F}})$-representations of $X$,
where we suppose that $\mathrm{char}({\mathbb{F}})=0$ and ${\mathbb{F}}$
contains the $\mathrm{l.c.d}(n,m,\ell)$-th root of unit. Then, all ${\rm
SL}_{2}({\mathbb{F}})$-representations of $X$ are already classified. As a
partial result, we mention the classification of such representations and the
acyclicity as follows:

######  Fact 8.2 ([Kit], [Wak]).

Let $\alpha\in{\mathbb{F}}^{\times}\setminus\\{\pm 1\\}$. Then, two matrices
$A=\begin{pmatrix}\alpha&0\\\ 0&{\alpha}^{-1}\\\
\end{pmatrix},B=\begin{pmatrix}x&y\\\ z&w\\\ \end{pmatrix}\in
SL(2;{\mathbb{F}})$ satisfy $(AB)^{n}=B^{m},(BA)^{n}=A^{\ell}$ if and only if
the following conditions hold.

  1. (IA)

$\alpha^{\ell}=\beta^{m}=\gamma^{n}\in\\{\pm 1\\},$

  2. (IB)

$x=\dfrac{\alpha^{-1}(\beta+\beta^{-1})-(\gamma+\gamma^{-1})}{\alpha^{-1}-\alpha},$
$w=\dfrac{(\gamma+\gamma^{-1})-\alpha(\beta+\beta^{-1})}{\alpha^{-1}-\alpha}.$

Here, $\beta,\beta^{-1}$ are the eigenvalues of $B$, and $\gamma,\gamma^{-1}$
are the eigenvalues of $AB$.

Any irreducible representation $\pi_{1}(X)\rightarrow{\rm
SL}_{2}({\mathbb{F}})$ is conjugate to the unitary representation
$\rho_{A,B}:\pi_{1}(X)\rightarrow{\rm SU}(2)$ such that $\rho_{A,B}(g)=A$ and
$\rho_{A,B}(h)=B$ for some matrices $A,B$ satisfying (IA)(IB). Furthermore,
for such a representation, the followings are known:

  1. (i)

If $\alpha^{\ell}=\beta^{m}=\gamma^{n}=-1$, then the complex
$C^{\rho_{A,B}}_{*}(X)$ is acyclic, and the torsion is given by

$T_{\rho_{A,B}}(M)=-\dfrac{4\alpha\beta\gamma}{(\alpha-1)^{2}(\beta-1)^{2}(\gamma-1)^{2}}\in{\mathbb{F}}^{\times}.$

  2. (ii)

If $\alpha^{\ell}=\beta^{m}=\gamma^{n}=1$, then $C^{\rho_{A,B}}_{*}(X)$ is not
acyclic.

Thus, as the remaining case, we now compute the refined torsion in case (ii).

######  Proposition 8.3.

Let $d$ be $\mathrm{l.c.d}(n,m,\ell)$, and ${\mathbb{F}}$ be the cyclotomic
field ${\mathbb{Q}}(\sqrt{-1}/2\pi d)$. Consider the unitary representation
$\rho_{A,B}:\pi_{1}(X)\rightarrow{\rm SU}(2)$ satisfying (ii) as above, and a
Hermitian $\rho_{A,B}$-invariant bilinear form $\psi$ defined by
$\psi(v,w):={}^{t}\overline{v}w$ for $v,w\in{\mathbb{F}}^{2}$. Then,
$H^{0}_{\rho_{A,B}}(X;{\mathbb{F}}^{2})=H^{3}_{\rho_{A,B}}(X;{\mathbb{F}}^{2})=0$
and
$H^{1}_{\rho_{A,B}}(X;{\mathbb{F}}^{2})=H^{2}_{\rho_{A,B}}(X;{\mathbb{F}}^{2})={\mathbb{F}}^{2}$,
and the torsion in Theorem 3.1 is equal to $1$ in
${\mathbb{F}}/N({\mathbb{F}})$.

###### Proof.

By the procedure reported in Section 8.1, the complex
$C^{*}_{\rho_{A,B}}(X;{\mathbb{F}}^{2})$ can be written as

$C^{*}_{\rho_{A,B}}(X;{\mathbb{F}}^{2}):0\longrightarrow{\mathbb{F}}^{2}\xrightarrow{\small\begin{pmatrix}E-A\\\
E-B\\\
\end{pmatrix}}{\mathbb{F}}^{4}\overset{0}{\longrightarrow}{\mathbb{F}}^{4}\xrightarrow{(E-B,E-A)}{\mathbb{F}}^{2}\longrightarrow
0.$

Since $E-A$ and $E-B$ are invertible, we immediately have
$H^{0}_{\rho_{A,B}}(X;{\mathbb{F}}^{2})\cong
H^{3}_{\rho_{A,B}}(X;{\mathbb{F}}^{2})\cong 0$. Moreover, for a rational
number

$r:=\frac{1}{(1-\alpha)(1-\alpha^{-1})}+\frac{1}{(1-\beta)(1-\beta^{-1})}\in{\mathbb{Q}}^{\times},$

and two matrixes

$S:=\begin{pmatrix}(E-A^{-1})^{-1}\\\ -(E-B^{-1})^{-1}\end{pmatrix},\hskip
14.22636ptT:=\frac{1}{r}\begin{pmatrix}-(E-B)^{-1}\\\
(E-A)^{-1}\end{pmatrix}.$

we can verify that the first cohomology
$H^{1}_{\rho_{A,B}}(X;{\mathbb{F}}^{2})$ is generated by
$Sf^{(2)}_{1},Sf^{(2)}_{2}$, and the second one is generated by
$Tf^{(2)}_{1},Tf^{(2)}_{2}$. Furthermore, we can verify from (22) that the
Hermitian form (5) with $(m,i)=(1,1)$ with respect to the generators
$Sf^{(2)}_{1},Sf^{(2)}_{2}$ is represented by the diagonal matrix
$(1)\oplus(1)$. Thus, we can take the representative base of $B^{*}$ and
$H^{*}$ as

$\widetilde{{\mathbf{h}}}_{0}=\widetilde{{\mathbf{h}}}_{0}^{\rm
dual}=\emptyset,\hskip
14.22636pt\widetilde{{\mathbf{h}}}_{1}=\\{Sf^{(2)}_{1},Sf^{(2)}_{2}\\},\hskip
14.22636pt\widetilde{{\mathbf{h}}}_{1}^{\rm
dual}=\\{Tf^{(2)}_{1},Tf^{(2)}_{2}\\},$ (23)
$\widetilde{{\mathbf{b}}}_{1}=\\{f^{(2)}_{1},f^{(2)}_{2}\\},\hskip
14.22636pt\widetilde{{\mathbf{b}}}_{2}=\widetilde{{\mathbf{b}}}_{4}=\emptyset,\hskip
14.22636pt\widetilde{{\mathbf{b}}}_{3}=\\{f^{(4)}_{1},f^{(4)}_{2}\\}.\hskip
14.22636pt$

Based on the definition of the torsion, we can compute it as

$\begin{split}\tau^{0}_{\rho_{A,B}}(X,\\{{\mathbf{h}}_{1},{\mathbf{h}}_{1}^{\rm
dual}\\})&=r^{2}\left((1-\alpha)(1-\alpha^{-1})+(1-\beta)(1-\beta^{-1})\right)^{2}\\\
&=1\in{\mathbb{F}}^{\times}/\langle
N({\mathbb{F}}),\mathop{\mathrm{det}}\nolimits(\pi_{1}(X))\rangle.\end{split}$

∎

Proposition 8.3 suggests that the torsion modulo $N({\mathbb{F}})$ is trivial;
however, as in Proposition 7.2, we focus on the case $b_{1}(X)\geq 1$ and
compute the refinement of the torsion. To this end, we need to detect the
condition that $H_{1}(X;{\mathbb{Z}})$ contains free submodules:

######  Lemma 8.4.

$H^{1}(X;{\mathbb{Z}})\cong{\mathbb{Z}}$ if and only if there exist
$a,b,k\in{\mathbb{N}}$ such that $a$ is relatively prime to $b$ and
$\ell=b(a+b)k,m=a(a+b)k$, $n=abk$.

###### Proof.

Considering the presentation of $\pi_{1}(X)=\langle g,h\ |\ r,s\rangle$, we
can verify that $H^{1}(X;{\mathbb{Z}})\cong{\mathbb{Z}}$ if and only if
$\ell,m,$ and $n$ satisfy equation $m^{-1}+\ell^{-1}=n^{-1}$. Moreover, by an
elementary argument, we can see that the equation implies
$\ell=a(a+b)k,m=b(a+b)k$ and $n=abk$ for some $k\in{\mathbb{N}}$ and
$a,b\in{\mathbb{Z}}$ are required. ∎

By this lemma, it suffices to consider $,a,b,\in{\mathbb{Z}}$ such that
$\ell=a(a+b)k,m=b(a+b)k$, and $n=abk$, as in Lemma 8.4. Then, the epimorphism
$\alpha_{a,b}:\pi_{1}(X)\rightarrow{\mathbb{Z}}$ by $\alpha_{a,b}(g)=a$ and
$\alpha_{a,b}(h)=b$ is a generator of
$H^{1}(X;{\mathbb{Z}})\cong{\mathbb{Z}}$. We now confirm the admissibility in
Proposition 7.3.

######  Lemma 8.5.

Suppose $H^{1}(X;{\mathbb{Z}})\cong{\mathbb{Z}}$. Consider the unitary
representation $\rho_{A,B}:\pi_{1}(X)\rightarrow{\rm SU}(2)$ satisfying (ii)
above, and the Hermitian $\rho_{A,B}$-invariant bilinear form
$\psi(v,w):={}^{t}\overline{v}w$ for $v,w\in{\mathbb{F}}^{2}$. Then,
$H^{0}\oplus H^{3}(X;{\mathbb{F}}^{2})=0$ and
$H^{1}_{\rho_{A,B}}(X;{\mathbb{F}}^{2})=H^{2}_{\rho_{A,B}}(X;{\mathbb{F}}^{2})={\mathbb{F}}^{2}$;
thus, the triple $(\rho_{A,B},\psi,\Sigma_{\alpha_{a,b}})$ is admissible.

###### Proof.

Under the same notation as in Theorem 8.3, $\Psi$ denotes the matrix of the
anti-Hermitian form (15) with $(\rho_{A,B},\psi,\Sigma_{\alpha_{a,b}})$
related to the basis $\widetilde{{\mathbf{h}}}_{1}$. Then, with the help of a
computer program, we can verify that $\Psi$ is equal to an anti-Hermitian
matrix

$\begin{split}&\dfrac{1}{ab(a+b)k}(E-A)^{-1}\left((E-A)^{-1}-(E-AB)^{-1}\right)(E-A^{-1})^{-1}\\\
&\quad+(E-A)^{-1}(B-A^{-1})^{-1}(E-B^{-1})^{-1}+(E-B)^{-1}(B^{-1}-A)^{-1}(E-A^{-1})^{-1}\\\
&\quad+(E-B)^{-1}\left((E-B)^{-1}-(E-BA)^{-1}\right)(E-B^{-1})^{-1},\end{split}$

and that $\Psi$ has eigenvalues $\sqrt{-1}u^{2}$ and $-\sqrt{-1}u^{2}$. Here,
$u^{2}=b(a+b)kr^{2}/(\lvert 1-\alpha\rvert\,\lvert 1-\beta\rvert\,\lvert
1-\gamma\rvert)\in{\mathbb{F}}$. Thus,
$(\rho_{A,B},\psi,\Sigma_{\alpha_{a,b}})$ is admissible as desired. ∎

Thus, by Lemma 8.5 and using $u$ above, we can choose a unitary matrix
$R_{a,b}\in\mathrm{SU}(2)$ such that

$\overline{(uR)}^{\mathrm{transpose}}M(uR)=\sqrt{-1}\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}.$

Here, $M$ represents the matrix presentation of the pairing in (15).

######  Example 8.6.

We can explicitly describe the matrix $R_{a,b}$; however, the explicit
description is generally complicated. For instance, if $\alpha\beta=1$ and

$B=\left(\begin{array}[]{cc}\cos\theta&-\sin\theta\\\ \sin\theta&\cos\theta\\\
\end{array}\right)\left(\begin{array}[]{cc}\frac{1}{\alpha}&0\\\ 0&\alpha\\\
\end{array}\right)\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\\\ \end{array}\right)$

for some $0<\theta<\frac{\pi}{2}$, then
$R_{a,b}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}-\sqrt{1-\sin\theta}&-\sqrt{1+\sin\theta}\\\
\sqrt{1+\sin\theta}&-\sqrt{1-\sin\theta}\\\ \end{array}\right),$ and hence
$\mathrm{det}R_{a,b}=1$.

######  Proposition 8.7.

Under the same notation in Lemma 8.5, the torsion in Theorem 7.2 is equal to

$a^{2}b^{2}(a+b)^{2}k^{2}\dfrac{(\lvert 1-\alpha\rvert^{2}+\lvert
1-\beta\rvert^{2})^{4}}{\lvert 1-\alpha\rvert^{6}\,\lvert
1-\beta\rvert^{6}\,\lvert
1-\gamma\rvert^{2}}\mathrm{det}R_{a,b}\in{\mathbb{C}}^{\times}/\\{\pm 1\\}.$

###### Proof.

Let us replace
$\widetilde{{\mathbf{h}}}_{0},\widetilde{{\mathbf{h}}}_{1},\widetilde{{\mathbf{h}}}_{2}$
and $\widetilde{{\mathbf{h}}}_{3}$ in (23) with

$\widetilde{{}^{\backprime}{\mathbf{h}}}_{0}=\widetilde{{}^{\backprime}{\mathbf{h}}}_{0}^{\rm
dual}=\emptyset,\hskip
14.22636pt\widetilde{{}^{\backprime}{\mathbf{h}}}_{1}=(uSR_{a,b}f^{(2)}_{1},uSR_{a,b}f^{(2)}_{2}),\hskip
14.22636pt\widetilde{{}^{\backprime}{\mathbf{h}}}_{1}^{\rm
dual}=\left(\frac{r}{u}TR_{a,b}f^{(2)}_{1},\frac{r}{u}TR_{a,b}f^{(2)}_{2}\right),$

respectively. Then, we can check that
$\widetilde{{}^{\backprime}{\mathbf{h}}}_{1}$ is an orthogonal basis with
respect to the pairing (15). Hence, from the definition of torsion, we get

$\begin{split}\tau^{0}_{\rho_{A,B}}(X,\\{^{\backprime}{\mathbf{h}}_{1},^{\backprime}{\mathbf{h}}_{1}^{\rm
dual})&=\dfrac{u^{4}}{r^{2}}(\lvert 1-\alpha\rvert^{2}\,+\lvert
1-\beta\rvert^{2}\,)^{2}\mathrm{det}R_{a,b}\\\
&=a^{2}b^{2}(a+b)^{2}k^{2}\dfrac{(\lvert 1-\alpha\rvert^{2}+\lvert
1-\beta\rvert^{2})^{4}}{\lvert 1-\alpha\rvert^{6}\,\lvert
1-\beta\rvert^{6}\,\lvert
1-\gamma\rvert^{2}}\mathrm{det}R_{a,b}\in{\mathbb{F}}^{\times}.\end{split}$

∎

###  8.3 Computation 2; Torus bundles

When $G={\rm SU}(2)$ and ${\mathbb{F}}={\mathbb{C}}$, we compute the adjoint
torsions of torus bundles over $S^{1}$.

First, we review torus bundles over $S^{1}$. Let $T^{2}$ be the torus. We fix
integers $\alpha,\beta,\gamma,\delta$ such that $\alpha\delta-\beta\gamma=\pm
1$ and denote the matrix $\begin{pmatrix}\alpha&\gamma\\\ \beta&\delta\\\
\end{pmatrix}$ by $A$. Then, we have a homeomorphism $f:T^{2}\rightarrow
T^{2}$, which induces the homomorphism
$f_{*}:\pi_{1}(T^{2})\rightarrow\pi_{1}(T^{2})$ as matrix $A.$ The mapping
torus $T_{f}$ is the quotient space of $T^{2}\times[0,1]$ subject to the
relation $(y,0)\sim(f(y),1)$ for any $y\in T^{2}$. The homeomorphism type of
$T_{f}$ depends on the mapping class of $f$.

We construct an identity of the torus bundle. To this end, we choose a
generating set $\\{a,b\\}$ of $\pi_{1}(T^{2})$, which gives the isomorphism
$\pi_{1}(T^{2})\cong\langle a,b\mid aba^{-1}b^{-1}\rangle.$ Following a van
Kampen argument, we can verify the presentation of $\pi_{1}(T_{f})$ as

$\langle\ a,b,t\mid ta^{\alpha}b^{\beta}t^{-1}a^{-1},\
ta^{\gamma}b^{\delta}t^{-1}b^{-1},aba^{-1}b^{-1}\ \rangle.$ (24)

Suppose some $q_{f}$ in the free group $\langle a,b\mid\rangle$ satisfying

$f_{*}(aba^{-1}b^{-1})=q_{f}(aba^{-1}b^{-1})q_{f}^{-1}\in\langle
a,b\mid\rangle,$ (25)

and consider the identity of the form

$W=\rho_{1}(a\rho_{2}a^{-1})(aba^{-1})\rho_{1}^{-1}(aba^{-1})^{-1}(aba^{-1}b^{-1})\rho_{2}^{-1}((aba^{-1}b^{-1}))^{-1}\rho_{3}tq_{f}\rho_{3}^{-1}q_{f}^{-1}t^{-1}.$

Then, $W$ is an identity satisfying the condition in Theorem 8.1; see, e.g.,
[Nos, Theorem 3.1].

As is known [Kit], the ${\rm SU}(2)$-character variety $R_{{\rm
SU}(2)}(T_{f})$ as a subset of
$\mathop{\mathrm{Hom}}\nolimits(\pi_{1}(T_{f}),{\rm SU}(2))$/conj is
homomorphic to the set

$\\{(u,v)\in{\mathbb{C}}^{2}\mid u^{\alpha+1}v^{\beta}=1,\
u^{\gamma}v^{\delta+1}=1,\ (u^{2},v^{2})\neq(1,1),\ |u|=|v|=1\\},$ (26)

almost everywhere. We may suppose the open set $R_{{\rm SU}(2)}(T_{f})^{o}$ to
be the set (26). In particular, $R(T_{f})$ is of infinite order if and only if
$\alpha=\delta=-1$ and $\beta\gamma=0$, and if and only if the cohomology
$H^{1}(T_{f};\mathfrak{su}_{2})$ is not zero. We have interests in acyclic
cases, and therefore, we focus on such a case and assume $\alpha=\delta=-1$
and $\gamma=0$. For $(u,v)$ in the set (26), we define a representation
$\rho_{u,v}:\pi_{1}(T_{f})\rightarrow{\rm SU}(2)$ by

$\rho_{u,v}(a)=\begin{pmatrix}u&0\\\ 0&u^{-1}\\\ \end{pmatrix},\ \ \
\rho_{u,v}(b)=\begin{pmatrix}v&0\\\ 0&v^{-1}\\\ \end{pmatrix},\ \ \
\rho_{u,v}(t)=\begin{pmatrix}0&1\\\ -1&0\\\ \end{pmatrix}.$

Then, the correspondence $(u,v)\mapsto\rho_{u,v}$ results in an injection to
an open dense subset $R_{{\rm SU}(2)}(T_{f})$.

Under a canonical identification
$\mathfrak{g}\otimes{\mathbb{C}}^{3}\cong\mathbb{C}^{3}$, the adjoint action
is obtained by

$\mathrm{Ad}\rho_{u,v}(a)=\begin{pmatrix}u^{2}&0&0\\\ 0&1&0\\\ 0&0&u^{-2}\\\
\end{pmatrix},\ \ \ \mathrm{Ad}\rho_{u,v}(b)=\begin{pmatrix}v^{2}&0&0\\\
0&1&0\\\ 0&0&v^{-2}\\\ \end{pmatrix},\ \ \
\mathrm{Ad}\rho_{u,v}(t)=\begin{pmatrix}0&0&-1\\\ 0&-1&0\\\ -1&0&0\\\
\end{pmatrix}.$

For each $\rho_{a,b}$, we now compute the torsion of $T_{f}$ in Theorem 3.1.

######  Proposition 8.8.

Let $G={\rm SU}(2)$. Suppose $\alpha=\delta=-1,\gamma=0$, and $\beta\neq 0.$
Let $\psi$ be the Killing form. Then, for the representation $\rho_{u,v}$, the
cohomology $H^{0}\oplus H^{3}(T_{f};{\mathbb{C}}^{2})$ is trivial and
$H^{1}(T_{f};{\mathbb{C}}^{2})\cong
H^{2}(T_{f};{\mathbb{C}}^{2})\cong{\mathbb{C}}$, and the torsion
$\tau^{0}_{\rho_{u,v}}(X,\\{{\mathbf{h}}_{1},{\mathbf{h}}_{1}^{\rm
dual}\\}){\mathbf{h}}_{1}^{\otimes 2}$ in Theorem 3.1 is
$-4/\beta{\mathbf{h}}_{1}^{\otimes 2}$. In addition, the volume in Definition
4.2 is $4\beta^{1/2}\pi$.

###### Proof.

Based on the procedure defined in Section 8.1,
$C^{*}_{\mathrm{Ad}\rho_{u,v}}(X;{\mathbb{C}}^{3})$ is described as

$C^{*}_{\mathrm{Ad}\rho_{u,v}}(X;{\mathbb{C}}^{3}):0\longrightarrow{\mathbb{C}}^{3}\overset{\partial^{0}}{\longrightarrow}{\mathbb{C}}^{9}\overset{\partial^{1}}{\longrightarrow}{\mathbb{C}}^{9}\overset{\partial^{2}}{\longrightarrow}{\mathbb{C}}^{3}\longrightarrow
0,$

where $\partial^{0},\partial^{1}$, and $\partial^{2}$ are defined by

$\partial^{0}(x)=\begin{pmatrix}1-u^{2}&0&0\\\ 0&0&0\\\ 0&0&1-u^{-2}\\\
1-v^{2}&0&0\\\ 0&0&0\\\ 0&0&1-v^{-2}\\\ 1&0&1\\\ 0&2&0\\\
1&0&1\end{pmatrix}x,$
$\partial^{1}(y)=\begin{pmatrix}-1&0&u^{2}&0&0&0&1-u^{2}&0&0\\\
0&0&0&0&-\beta&0&0&0&0\\\ u^{-2}&0&-1&0&0&0&0&0&1-u^{-2}\\\
0&0&0&-1&0&v^{2}&1-v^{2}&0&0\\\ 0&0&0&0&0&0&0&0&0\\\
0&0&0&v^{-2}&0&-1&0&0&1-v^{-2}\\\ 1-v^{2}&0&0&u^{2}-1&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0\\\ 0&0&1-v^{-2}&0&0&u^{-2}-1&0&0&0\end{pmatrix}y,$
$\partial^{2}(y)=\begin{pmatrix}1-v^{2}&0&0&u^{2}-1&0&0&1&0&u^{2}v^{2}\\\
0&0&0&0&0&0&0&2&0\\\
0&0&1-v^{-2}&0&0&u^{-2}-1&u^{-2}v^{-2}&0&1\end{pmatrix}y.$

for $x\in{\mathbb{C}}^{3}$ and $y\in{\mathbb{C}}^{9}$. Accordingly, we can
take bases of the forms

$\widetilde{{\mathbf{h}}}_{0}=\widetilde{{\mathbf{h}}}_{0}^{\rm
dual}=\emptyset,\hskip
14.22636pt\widetilde{{\mathbf{h}}}_{1}=\\{f^{(9)}_{2}\\},\hskip
14.22636pt\widetilde{{\mathbf{h}}}_{2}=\\{f^{(9)}_{2}\\},$ (27)
$\tilde{\textbf{b}}_{1}=\\{f^{(3)}_{1},f^{(3)}_{2},f^{(3)}_{3}\\},\hskip
14.22636pt\tilde{\textbf{b}}_{2}=\\{f^{(9)}_{1},f^{(9)}_{3},f^{(9)}_{5},f^{(9)}_{7},f^{(9)}_{9}\\},\hskip
14.22636pt\tilde{\textbf{b}}_{3}=\\{f^{(9)}_{1},f^{(9)}_{3},f^{(9)}_{8}\\}.$

We can confirm that the matrix of the Hermitian form (5) with respect to the
bases $\widetilde{{\mathbf{h}}}_{1}$ and $\widetilde{{\mathbf{h}}}_{2}$
coincides with $1$. Furthermore, since $q_{f}$ in (25) equals $a^{-1}b^{-1}$,
we can compute the pairing in (15); i.e., we can confirm
$\widetilde{{\mathbf{h}}}_{1}^{\text{dual}}=\widetilde{{\mathbf{h}}}_{2}$.
Hence, we can compute the torsion
$\tau^{0}_{\rho_{u,v}}(T_{f},\\{{\mathbf{h}}_{1},{\mathbf{h}}_{1}^{\rm
dual}\\})$ as $-4/\beta$ from the definition of torsions.

Finally, to compute the volume, we note from (26) that the set $R_{{\rm
SU}(2)}(T_{f})$ is homomorphic to $\sqcup^{\beta}S^{1}$ almost everywhere.
Furthermore, by a dual argument, the torsion
$\tau^{0}_{\rho_{u,v}}(T_{f},\\{{\mathbf{h}}_{2},{\mathbf{h}}_{2}^{\rm
dual}\\})^{1/2}$ as $2/\beta^{1/2}{\mathbf{h}}_{2}^{\otimes}\in
H^{2}(T_{f};\mathfrak{su}(2))$ is a constant form on $R_{{\rm SU}(2)}(T_{f})$.
Thus, the volume is $(2/\beta^{1/2})\times 2\pi\beta=4\beta^{1/2}\pi.$ ∎

However, the torsion $-\beta/4\in{\mathbb{R}}$ modulo
$\mathbb{C}^{\times}/N(\mathbb{C})\cong S^{1}$ is $-1$ trivially. Thus, it is
worth considering the refinements in Section 7.

######  Proposition 8.9.

Under the same assumptions of Proposition 8.8, for any
$\mathrm{Ad}\rho_{u,v}$-invariant bilinear form $\psi$ and surjection
$\alpha:H_{1}(T_{f})\rightarrow{\mathbb{Z}}$, the pair
$(\mathrm{Ad}\rho_{u,v},\psi,\Sigma_{\alpha})$ is not admissible.

###### Proof.

There are only two surjections
$\alpha_{+},\alpha_{-}:H_{1}(T_{f})\rightarrow{\mathbb{Z}}$ that satisfy
$\alpha_{+}(t)=-\alpha_{-}(t)=1$ and $\alpha_{\pm}(a)=\alpha_{\pm}(b)=0$. It
follows easily that $-\Sigma_{\alpha_{+}}=\Sigma_{\alpha_{-}}=(1\otimes
f^{(3)}_{3})\in{\mathbb{Z}}\otimes C_{1}(\widetilde{X};{\mathbb{Z}})$. In
addition, $(f^{(9)}_{2}\otimes
f^{(9)}_{2})(\Upsilon(aba^{-1}b^{-1}))=0\in{\mathbb{C}}^{3}\otimes{\mathbb{C}}^{3}.$
Hence, $\psi_{*}((f^{(9)}_{2}\smile f^{(9)}_{2})\cap\Sigma_{\alpha_{\pm}})=0$
for any $\mathrm{Ad}\rho_{u,v}$-invariant bilinear form $\psi$, which implies
that the pairs $(\mathrm{Ad}\rho_{u,v},\psi,\Sigma_{\alpha_{\pm}})$ are not
admissible. ∎

Thus, we cannot apply the torsion to Proposition 7.2. In contrast, the
refinement in Proposition 7.3 is not trivial as follows:

######  Proposition 8.10.

Let $\Sigma$ be a fiber of $T_{f}$ homomorphic to $T^{2}$. Then, the inclusion
$\Sigma\hookrightarrow T_{f}$ induces the isomorphism
$H^{2}(T_{f};{\mathbb{C}}^{n})\cong
H^{2}(\Sigma_{\alpha};{\mathbb{C}}^{n})\cong{\mathbb{C}}$ as in Proposition
7.3. Thus, the refined torsion (17) is equal to $|\beta|/4$ in
$\mathbb{C}^{\times}/\\{\pm 1\\}.$

The proof is obvious by the description of $\widetilde{{\mathbf{h}}}_{2}$ in
(27).

### Acknowledgments

The authors sincerely express their gratitude to Yoshiakzu Yamaguchi for the
valuable comments.

##  Appendix A Proof of Proposition 3.2

Hereafter, we denote the local coefficient module ${\mathbb{F}}^{n}$ by $A$
and the dual of the conjugate representation
$\overline{\mathrm{Hom}(A;{\mathbb{F}})}$ by $A^{*}$. We have an isomorphism
$\mathcal{D}:A\cong A^{*}$ of local coefficients by the correspondence
$a\mapsto\psi(\bullet,a)$. We note that $(A^{*})^{*}\cong A$ and $\mathcal{D}$
yield an isomorphism $\mathcal{D}_{*}:H^{*}(X;A)\cong H^{*}(X;A^{*}).$

###### Proof of Proposition 3.2.

Since Char${\mathbb{F}}=0$, we have
$\mathrm{Ext}^{1}(H^{p}(X;A);{\mathbb{F}})=0$; therefore, by the universal
coefficient theorem, the Kronecker product gives an isomorphism

$\mathcal{K}:H^{p}(X;A^{*})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Hom}(H_{p}(X;(A^{*})^{*});{\mathbb{F}})\cong\mathrm{Hom}(H_{p}(X;A),{\mathbb{F}}),$

which implies $\mathrm{dim}H^{n}(X;A^{*})=\mathrm{dim}H_{n}(X;A)$. Thus,
Poincaré duality leads to

$\mathop{\mathrm{dim}}\nolimits H^{p}(M;A)=\mathop{\mathrm{dim}}\nolimits
H_{2m+1-p}(M;A)=\mathop{\mathrm{dim}}\nolimits H^{2m+1-p}(M;A).$

Next, to show the nondegeneracy of the pairing (5), we examine the composite
of $\psi$ and the cap-product

$\cap_{\psi}:=\psi\circ\cap:H_{p}(X;A)\otimes
H^{p}(X;A)\longrightarrow{\mathbb{F}}.$ (28)

Furthermore, we consider the coupling

$H_{p}(X;A)\otimes H^{p}(X;A)\longrightarrow{\mathbb{F}};\ \ \ a\otimes
f\longmapsto\mathcal{K}(\mathcal{D}_{*}(f))(a).$ (29)

The definitions of the cap-product, $\mathcal{D}_{*}$, and $\mathcal{K}$
suggest that the maps (28) and (29) are equal up to signs. Thus, the Kronecker
delta is nondegenerate, and so is the map $\cap_{\psi}$.

Finally, we will complete the proof. Suppose $a\in H^{p}(X;A)$ such that
$a\smile_{\psi}b=0$ for any $b\in H^{2m+1-p}(X;A)$. It suffices to show that
$a=0$. Note that

$a\smile_{\psi}b=\psi([M]\smile(a\smile b))=\psi(([M]\cap a)\cap
b)=\cap_{\psi}([M]\cap a,b).$

Since $\cap_{\psi}$ is nondegenerate as above, $[M]\cap a=0$. Hence, Poincaré
duality deduces $a=0$ as desired. ∎

##  Appendix B Comparison with the works of Faber and Turaev

As mentioned in Section 7, we show that the refined torsion in Theorem 3.1 is
a modification of the Poincaré-Reidemeister (scalar) product [Far, FT].

Throughout this appendix, as in §7, we suppose a closed manifold $X$ of
dimension $2m+1$ with an ${\rm SL}$-representation
$\rho:\pi_{1}(X)\rightarrow{\rm SL}_{n}({\mathbb{F}})$ and an (anti-)Hermitian
$\rho$-invariant bilinear form
$\psi:{\mathbb{F}}^{n}\times{\mathbb{F}}^{n}\rightarrow{\mathbb{F}}$. Here
${\mathbb{F}}$ represents an arbitrary field of characteristic zero444In the
original papers [Far, FT], the authors considered only a flat vector bundle
with unimodulalirty and a Poincaré duality, where ${\mathbb{F}}$ is either
${\mathbb{R}}$ or $\mathbb{C}$. However, the duality in [Far] contained an
ambiguous concept; thus, to clarify the duality as in Proposition 3.2, this
paper supposes such a $\psi$ and deals with fields of characteristic zero. .
Tthe torsion defined in this paper is valued in
${\mathbb{F}}^{\times}/N({\mathbb{F}})$; see (6). Here, let us note the
following lemma:

######  Lemma B.1.

Let $W$ be a 1-dimensional ${\mathbb{F}}$-space with a basis $w_{0}\in W.$
Then, the set of the isomorphism classes of nontrivial Hermitian bilinear
forms $\phi:W\times W\rightarrow{\mathbb{F}}$ is in a 1:1-correspondence with
${\mathbb{F}}^{\times}/N({\mathbb{F}})$. Here, the correspondence is given by
$\phi\mapsto[\phi(w_{0},w_{0})]$ as the discriminant, and it is independent of
the choice of the basis $w_{0}$.

Thus, it is sensible to study the torsion as a Hermitian bilinear form as
follows.

To this end, let us establish some terminology. For a finite dimensional
vector space $W$, let $\mathop{\mathrm{det}}\nolimits(W)$ denote the exterior
product $\Lambda^{\mathop{\mathrm{dim}}\nolimits W}(W)$, and
$\mathop{\mathrm{det}}\nolimits(W)^{-1}$ denote the dual
$\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{F}}}(\mathop{\mathrm{det}}\nolimits(W),{\mathbb{F}})$.
Here, if $W=\\{0\\}$, then
$\mathop{\mathrm{det}}\nolimits(W)=\mathop{\mathrm{det}}\nolimits(W)^{-1}={\mathbb{F}}$
by definition. For a finite-dimensional graded vector space $V=V_{0}\oplus
V_{1}\oplus\cdots\oplus V_{k}$, the determinant line of $V$ is defined as

$\mathop{\mathrm{det}}\nolimits
V=\mathop{\mathrm{det}}\nolimits(V_{0})\otimes\mathop{\mathrm{det}}\nolimits(V_{1})^{-1}\otimes\mathop{\mathrm{det}}\nolimits(V_{2})\otimes\cdots\otimes\mathop{\mathrm{det}}\nolimits(V_{k})^{(-1)^{k}}.$

For a basis $\mathbf{h}=(\mathbf{h}_{0},\dots,\mathbf{h}_{2m+1})$ of
$H^{*}(X;{\mathbb{F}}^{n})$, we write $\mathbf{h}^{\otimes}$ for
$\mathbf{h}_{0}\otimes\mathbf{h}_{1}^{-1}\otimes\mathbf{h}_{2}\otimes\cdots\otimes\mathbf{h}_{2m+1}^{-1}\in\mathop{\mathrm{det}}\nolimits(H^{*}(X;{\mathbb{F}}^{n}))$.
Then, using the refined torsion $\tau_{\rho}^{0}$ in (4), we define
$\tau_{\rho}(X)\in\mathop{\mathrm{det}}\nolimits H^{*}(X;{\mathbb{F}}^{n})$ to
be $\tau_{\rho}^{0}(X,\mathbf{h})\mathbf{h}^{\otimes}$. Here, $\tau_{\rho}(X)$
is known to be independent of the choices of $\mathbf{h}$.

We set the conjugate representations $\bar{\rho}:\pi_{1}(X)\rightarrow{\rm
SL}_{n}({\mathbb{F}})$ defined by $\bar{\rho}(g)=\overline{\rho(g)}$ for
$g\in\pi_{1}(X)$, and the associated cohomology by
$H^{*}(X;({\mathbb{F}}^{n})^{*})$. Further, we consider the direct sum
$\rho\oplus\bar{\rho}:\pi_{1}(X)\rightarrow{\rm SL}_{2n}({\mathbb{F}})$.
Recall from Proposition 3.2 the isomorphism $D$ as a Poincaré duality, which
induces isomorphism $D:\mathop{\mathrm{det}}\nolimits
H^{*}(X;{\mathbb{F}}^{n})\rightarrow\mathop{\mathrm{det}}\nolimits
H^{*}(X;({\mathbb{F}}^{n})^{*})$. Define

$\mu^{\bullet}:\mathop{\mathrm{det}}\nolimits
H^{*}(X;{\mathbb{F}}^{n})\otimes\mathop{\mathrm{det}}\nolimits
H^{*}(X;({\mathbb{F}}^{n})^{*})\longrightarrow\mathop{\mathrm{det}}\nolimits
H^{*}(X;{\mathbb{F}}^{n}\oplus({\mathbb{F}}^{n})^{*}),$

by $\mu^{\bullet}(v_{1}\wedge\cdots\wedge v_{m},w_{1}\wedge\cdots\wedge
w_{m})=v_{1}\wedge\cdots\wedge v_{m}\wedge w_{1}\wedge\cdots\wedge w_{m}$. As
in Section 4.7 in [Far] and Section 9 in [FT], the Poincaré-Reidemeister
(scalar) product is defined as a Hermitian product by setting

${\rm PR}:\mathop{\mathrm{det}}\nolimits
H^{*}(X;{\mathbb{F}}^{n})\times\mathop{\mathrm{det}}\nolimits
H^{*}(X;{\mathbb{F}}^{n})\longrightarrow{\mathbb{F}}^{\times};(\alpha,\beta)\longmapsto\frac{\mu^{\bullet}(\alpha\otimes
D(\beta))}{\tau_{\rho\oplus\bar{\rho}}(X)}.$ (30)

However, we show that this product is trivial in the sense of Lemma B.1. For
this, let $H^{\rm half}(X;{\mathbb{F}}^{n})$ be the graded vector space
$\oplus_{j=0}^{m}H^{j}(X;{\mathbb{F}}^{n})$. Similarly, the Poincaré duality
induces an isomorphism $D:\mathop{\mathrm{det}}\nolimits H^{\rm
half}\rightarrow\mathop{\mathrm{det}}\nolimits(\oplus_{j=m+1}^{2m+1}H^{j}(X;{\mathbb{F}}^{n}))$.
Then, by definition, we can easily show

${\rm
PR}(\tau_{\rho}^{0}(X,\mathbf{h}^{\prime})\mathbf{h}^{\prime},\tau_{\rho}^{0}(X,\mathbf{h}^{\prime})\mathbf{h}^{\prime})=1\in{\mathbb{F}},$

where $\mathbf{h}^{\prime}=\mathbf{h}^{\rm half}\otimes D(\mathbf{h}^{\rm
half})^{-1}$ and $\mathbf{h}^{\rm half}\in H^{\rm half}(X;{\mathbb{F}}^{n})$
is arbitrary. Hence, by Lemma B.1, the bilinear isomorphism class of ${\rm
PR}$ is trivial, as required.

We consider a modification of the PR product, and provide a comparison with
the refined torsion. To be precise, we define a half PR product by the
Hermitian product

${\rm PR}^{\rm half}:\mathop{\mathrm{det}}\nolimits H^{\rm
half}(X;{\mathbb{F}}^{n})\times\mathop{\mathrm{det}}\nolimits H^{\rm
half}(X;{\mathbb{F}}^{n})\longrightarrow{\mathbb{F}}^{\times};(\alpha,\beta)\longmapsto\frac{\mu^{\bullet}(\alpha\otimes
D(\beta))}{\tau_{\rho}(X)}.$ (31)

By definition, we can check that, for any $\mathbf{h}^{\rm half}\in H^{\rm
half}(X;{\mathbb{F}}^{n})$, ${\rm PR}^{\rm half}(\mathbf{h}^{\rm
half},\mathbf{h}^{\rm half})$ is equal to the inverse of the refined torsion
in (6). In conclusion, it follows from Lemma B.1 that the bilinear form
associated with the refined torsion in (6) is equal to the half PR product
(31), as a modification of the original ${\rm PR}$ product.

## References

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Department of Mathematics, Tokyo Institute of Technology 2-12-1 Ookayama,
Meguro-ku Tokyo 152-8551 Japan


t_{j}{(i)}\left\\{H(\mathbb{X}_{j}{(1:i)}+H(\mathbb{X}_{j}{(i+1:N_{j})})-H(\mathbb{X}_{j})\right\\}\right]\mathbbm{1}{\left\\{A_{j-1}^{c}\cap
A_{j}\cap A_{j+1}^{c}\right\\}}$
$\displaystyle=\sum_{j=2}^{m_{n}-1}\mathcal{IC}_{j}(H)\;.$

##### Explanation for (3.11).

We analyse ${\rm{SB}}_{j-1}(H)$ on $A_{j-1}^{c}\cap A_{j}$. The index $i$
comes from the $(j-1)$th block: $(j-2)r_{n}+1\leqslant i\leqslant(j-1)r_{n}$.
A contribution to ${\rm{SB}}_{j-1}(H)$ comes from the jumps in block $j$ only.

  * (I-a)

At the beginning of the $(j-1)$th block: If $(j-1)r_{n}\leqslant
i+r_{n}-1<t_{j}(1)$, then $H\equiv 0$.

  * (I-b)

In the middle of the $(j-1)$th block: Let $\ell=1,\ldots,N_{j}-1$. If
$t_{j}(\ell)\leqslant i+r_{n}-1<t_{j}(\ell+1)$, then the contribution is
$H(u_{n}^{-1}\boldsymbol{X}_{i,i+r_{n}-1})=H(u_{n}^{-1}\boldsymbol{X}_{t_{j}(1),t_{j}(\ell)})=H(\mathbb{X}_{j}{(1:\ell)})$.
This happens for $\Delta t_{j}(\ell)=t_{j}(\ell+1)-t_{j}(\ell)$ of the
indices.

  * (I-c)

At the end of the $(j-1)$th block: If $t_{j}{(N_{j})}\leqslant
i+r_{n}-1<jr_{n}=t_{j}{(N_{j}+1)}$, then the contribution is
$H(u_{n}^{-1}\boldsymbol{X}_{i,i+r_{n}-1})=H(u_{n}^{-1}\boldsymbol{X}_{t_{j}(1),t_{j}{(N_{j})}})=H(\mathbb{X}_{j})$.
This happens for $jr_{n}-t_{j}{(N_{j})}=t_{j}{(N_{j}+1)}-t_{j}{(N_{j})}=\Delta
t_{j}{(N_{j})}$ of the indices.

In summary, on $A_{j-1}^{c}\cap A_{j}$, we have

$\displaystyle{\rm{SB}}_{j-1}(H)$ $\displaystyle=\sum_{\ell=1}^{N_{j}}\Delta
t_{j}(\ell)H(\mathbb{X}_{j}{(1:\ell)})\;.$

##### Explanation for (3.12).

We analyse ${\rm{SB}}_{j+1}(H)$ on $A_{j+1}\cap A_{j+2}^{c}$ (then (3.12)
follows by replacing $j$ with $j-1$.) The index $i$ comes from the $(j+1)$th
block: $jr_{n}+1\leqslant i\leqslant(j+1)r_{n}$. A contribution to
${\rm{SB}}_{j+1}(H)$ comes from the jumps in block $j+1$ only.

  * (II-a)

At the beginning of the $(j+1)$th block: If $jr_{n}+1\leqslant i\leqslant
t_{j+1}(1)$, then the contribution is $H(\mathbb{X}_{j+1}{(1:N_{j+1})})$. This
happens for $t_{j+1}(1)-jr_{n}=t_{j+1}(1)-t_{j+1}(0)=\Delta t_{j+1}(0)$ of the
indices.

  * (II-b)

In the middle of the $(j+1)$th block: Let $\ell=1,\ldots,N_{j+1}-1$. If
$t_{j+1}(\ell)<i\leqslant t_{j+1}(\ell+1)$, then the contribution is
$H(\mathbb{X}_{j+1}{(\ell+1:N_{j+1})})$. This happens for $\Delta
t_{j+1}(\ell)=t_{j+1}(\ell+1)-t_{j+1}(\ell)$ of the indices.

  * (II-c)

At the end of the $(j+1)$th block: If $t_{j+1}{(N_{j}+1)}<i$, then $H\equiv
0$.

In summary, on $A_{j+1}\cap A_{j+2}^{c}$, we have

$\displaystyle{\rm{SB}}_{j+1}(H)$
$\displaystyle=\sum_{\ell=0}^{N_{j+1}-1}\Delta
t_{j+1}(\ell)H(\mathbb{X}_{j+1}{(\ell+1:N_{j+1})})\;.$

##### Explanation for (3.20).

##### Contribution from ${\rm{SB}}_{j}(H)$ on $A_{j}\cap A_{j+1}$.

The index $i$ runs from $(j-1)r_{n}+1$ to $jr_{n}$. Contribution to
${\rm{SB}}_{j}$ comes from the jumps in both $j$th and $(j+1)$th block. This
contribution can be written in a more concise way, but the more detailed
decomposition below will be useful when combining
${\rm{SB}}_{j}(H)\mathbbm{1}{\left\\{A_{j}\cap A_{j+1}\right\\}}$ with other
terms.

Recall the convention $t_{j}(0)=(j-1)r_{n}$ and $t_{j}{(N_{j}+1)}=jr_{n}$.
Recall also the notation for adjacent blocks from Section 3.1.1.

Let $q=0,\ldots,N_{j}$ and $s=0,\ldots,N_{j+1}$. Let

$\displaystyle\mathcal{J}_{q,s}=\\{i=(j-1)r_{n}+1,\ldots,jr_{n}:t_{j}(q)<i\leqslant
t_{j}(q+1),t_{j+1}(s)\leqslant i+r_{n}-1<t_{j+1}(s+1)\\}\;.$

If $i\in\mathcal{J}_{q,s}$ then the contribution is
$H(\mathbb{X}_{j,j+1}{(q+1:N_{j}+s)})$.

  * (III-a)

If $t_{j}(q+1)\leqslant t_{j+1}(s)-r_{n}$, then $\mathcal{J}_{q,s}$ is empty
and there is no contribution.

  * (III-b)

If $t_{j}(q)<t_{j+1}(s)-r_{n}<t_{j}(q+1)\leqslant t_{j+1}(s+1)-r_{n}$, then we
have the contribution for $t_{j}(q+1)-(t_{j+1}(s)-r_{n})$ of the indices.

  * (III-c)

If $t_{j}(q)<t_{j+1}(s)-r_{n}<t_{j+1}(s+1)-r_{n}<t_{j}(q+1)$, then we have the
contribution for $t_{j+1}(s+1)-t_{j+1}(s)$ of the indices.

  * (III-d)

If $t_{j+1}(s)-r_{n}\leqslant t_{j}(q)<t_{j}(q+1)\leqslant
t_{j+1}(s+1)-r_{n}$, then we have the contribution for $t_{j}(q+1)-t_{j}(q)$
of the indices.

  * (III-e)

If $t_{j+1}(s)-r_{n}\leqslant t_{j}(q)<t_{j+1}(s+1)-r_{n}<t_{j}(q+1)$, then we
have the contribution for $t_{j+1}(s+1)-t_{j}(q)-r_{n}$ of the indices.

##### Joint contribution from ${\rm{SB}}_{j-1}$, ${\rm{SB}}_{j}$,
${\rm{SB}}_{j+1}$ on $A_{j-1}^{c}\cap A_{j}\cap A_{j+1}\cap A_{j+2}^{c}$.

We start by observing that the case $q=1,\ldots,N_{j}$, $s=0,\ldots,N_{j+1}-1$
in (III) simplifies:

  * •

The contribution $H(\mathbb{X}_{j,j+1}{(q+1:N_{j}+s)})$ can occur only if
$\mathcal{L}_{j,j+1}\geqslant r_{n}$.

The other cases of $q$ and $s$ require a precise analysis.

  * (IV-a)

The contribution (I-a), (I-b) remains the same. Note further that for
$\ell=1,\ldots,N_{j-1}$, $\Delta t_{j}(\ell)=\Delta t_{j,j+1}(\ell)$ and
$H(\mathbb{X}_{j}{(1:\ell)})=H(\mathbb{X}_{j,j+1}(1:\ell))$. In total, the
contribution from ${\rm{SB}}_{j-1}(H)$ in (I-a) and (I-b) can be written as

$\displaystyle\sum_{\ell=1}^{N_{j}-1}\Delta
t_{j,j+1}(\ell)H(\mathbb{X}_{j,j+1}(1:\ell))\;.$ (7.1)

  * (IV-b)

We analyse the contribution
$H(\mathbb{X}_{j})=H(\mathbb{X}_{j,j+1}(1:N_{j}))$. This stems from both
${\rm{SB}}_{j-1}$ and ${\rm{SB}}_{j}$. We combine (I-c) with (III) for
$q=s=0$.

    * –

(III-b) and (III-d) are duplicates in this case. If $t_{j}(1)\leqslant
t_{j+1}(1)-r_{n}$, we have the contribution for

$\displaystyle\underbrace{t_{j}(1)-(j-1)r_{n}}_{\textrm{from
(III)}}+\underbrace{(jr_{n}-t_{j}{(N_{j})})}_{\textrm{from
(I-c)}}=r_{n}-(t_{j}{(N_{j})}-t_{j}(1))$

of the indices.

    * –

(III-c) and (III-e) are duplicates in this case. If
$t_{j+1}(1)-r_{n}<t_{j}(1)$ we have the contribution for

$\displaystyle\underbrace{t_{j+1}(1)-jr_{n}}_{\textrm{from
(III)}}+\underbrace{(jr_{n}-t_{j}{(N_{j})})}_{\textrm{from
(I-c)}}=t_{j+1}(1)-t_{j}{(N_{j})}=\Delta t_{j,j+1}(N_{j})$

of the indices.

    * –

Thus, the contribution $H(\mathbb{X}_{j,j+1}(1:N_{j}))$ is for

$\displaystyle\Delta
t_{j,j+1}(N_{j})\mathbbm{1}{\left\\{t_{j+1}(1)-t_{j}(1)<r_{n}\right\\}}+\left(r_{n}-(t_{j}{(N_{j})}-t_{j}(1))\right)\mathbbm{1}{\left\\{t_{j+1}(1)-t_{j}(1)\geqslant
r_{n}\right\\}}$ (7.2)

of the indices. Note that the second part vanishes whenever the joint cluster
length fulfills $\mathcal{L}_{j,j+1}<r_{n}$.

  * (IV-c)

We take $q=0$ and $s=1,\ldots,N_{j+1}-1$ in (III). Then the contribution is
$H(\mathbb{X}_{j,j+1}{(1:N_{j}+s)})$. It stems from ${\rm{SB}}_{j}$ only. Set
$\ell=N_{j}+s$. With the re-numeration we have $\Delta t_{j+1}(s)=\Delta
t_{j,j+1}(\ell)$.

    * –

For (III-b): If $t_{j+1}(s)-r_{n}<t_{j}(1)\leqslant t_{j+1}(s+1)-r_{n}$, then
we have the contribution for $t_{j}(1)+r_{n}-t_{j+1}(s)\leqslant
t_{j+1}(s+1)-t_{j+1}(s)=\Delta t_{j+1}(s)$ of the indices.

    * –

Next, for (III-c), if $t_{j+1}(s+1)-r_{n}<t_{j}(1)$, then we have the
contribution for $t_{j+1}(s+1)-t_{j+1}(s)=\Delta t_{j+1}(s)$ of the indices.

    * –

Thus, $t_{j+1}(s+1)-t_{j}(1)\leqslant r_{n}$, the contribution occurs for at
most $\Delta t_{j+1}(s)$ of the indices.

    * –

The cases (III-d) and (III-e) are empty, since $t_{j+1}(s)-r_{n}\leqslant
t_{j}(0)=(j-1)r_{n}$ cannot happen.

    * –

Thus, the contribution for this part is at most

$\displaystyle\sum_{\ell=N_{j}+1}^{N_{j}+N_{j+1}-1}\Delta
t_{j,j+1}(\ell)H(\mathbb{X}_{j,j+1}(1:\ell))\;.$ (7.3)

Thus, whenever $\mathcal{L}_{j,j+1}<r_{n}$, (7.1), (7.2) and (7.3) give the
joint contribution from ${\rm{SB}}_{j-1}$ and ${\rm{SB}}_{j}$:

$\displaystyle\sum_{i=1}^{N_{j}+N_{j+1}-1}\Delta
t_{j,j+1}(i)H(\mathbb{X}_{j,j+1}{(1:i)})\;.$ (7.4)

Similar analysis applies to (II) and (III) combined, and we obtain

  * (IV-d)

the joint contribution from ${\rm{SB}}_{j}$ and ${\rm{SB}}_{j+1}$:

$\displaystyle\sum_{i=1}^{N_{j}+N_{j+1}-1}\Delta
t_{j,j+1}(i)H(\mathbb{X}_{j,j+1}{(i+1:N_{j}+N_{j+1})})\;.$ (7.5)

The case $q=0$ and $s=N_{j}$ needs a special care.

  * (IV-e)

We take $q=0$ and $s=N_{j+1}$ in (III). Then the contribution is
$H(\mathbb{X}_{j,j+1})$. It stems from ${\rm{SB}}_{j}$ only.

    * –

Start with (III-b). If $t_{j+1}{(N_{j+1})}-r_{n}<t_{j}(1)$ then we have the
contribution for $r_{n}-(t_{j+1}{(N_{j+1})}-t_{j}(1))$ of the indices.

    * –

The cases (III-c), (III-d) and (III-e) are empty.

    * –

Hence, the contribution is

$\displaystyle\left(r_{n}-\left(t_{j+1}{(N_{j+1})}-t_{j}(1)\right)\right)_{+}H(\mathbb{X}_{j,j+1})\;.$
(7.6)

We combine (7.6) with ${\rm{DB}}_{j,j+1}=r_{n}H(\mathbb{X}_{j,j+1})$ to get

$\displaystyle\left(r_{n}-\left(t_{j+1}{(N_{j+1})}-t_{j}(1)\right)\right)\mathbbm{1}{\left\\{t_{j+1}{(N_{j+1})}-t_{j}(1)<r_{n}\right\\}}H(\mathbb{X}_{j,j+1})-{\rm{DB}}_{j,j+1}$
$\displaystyle=-\left(t_{j+1}{(N_{j+1})}-t_{j}(1)\right)\mathbbm{1}{\left\\{t_{j+1}{(N_{j+1})}-t_{j}(1)<r_{n}\right\\}}H(\mathbb{X}_{j,j+1})$
(7.7)
$\displaystyle\phantom{=}-r_{n}\mathbbm{1}{\left\\{t_{j+1}{(N_{j+1})}-t_{j}(1)\geqslant
r_{n}\right\\}}H(\mathbb{X}_{j,j+1})\;.$

Now, with the re-numeration for the adjacent blocks:

$\displaystyle
t_{j+1}{(N_{j+1})}-t_{j}(1)=t_{j,j+1}{(N_{j}+N_{j+1})}-t_{j,j+1}(1)=\sum_{i=1}^{N_{j}+N_{j+1}-1}\Delta
t_{j,j+1}(i)\;.$

Combining this with (7.4)-(7.1) we have

$\displaystyle\big{[}{\rm{SB}}_{j-1}+{\rm{SB}}_{j}+{\rm{SB}}_{j+1}-{\rm{DB}}_{j,j+1}\big{]}\mathbbm{1}{\left\\{A_{j-1}^{c}\cap
A_{j}\cap A_{j+1}\cap A_{j+2}^{c}\right\\}}$
$\displaystyle=\sum_{i=1}^{N_{j}+N_{j+1}-1}\Delta
t_{j,j+1}(i)\left\\{H(\mathbb{X}_{j,j+1}{(1:i)})+H(\mathbb{X}_{j,j+1}{(i+1:N_{j}+N_{j+1})})-H(\mathbb{X}_{j,j+1})\right\\}$

whenever $\mathcal{L}_{j,j+1}<r_{n}$.

###  7.2 Extremal index case

##### Explanation for Example 3.2.2.

  * •

On $A_{j-1}^{c}\cap A_{j}$: for $i\in I_{j-1}$, we have
$H(\boldsymbol{X}_{i+1,i+r_{n}}/u_{n})=1$ if and only if $i+r_{n}-1\geqslant
t_{j}(1)$. Hence, ${\rm{SB}}_{j-1}(H)=(j-1)r_{n}-(t_{j}(1)-r_{n})$.

  * •

On $A_{j}\cap A_{j+1}^{c}$: for $i\in I_{j}$, we have
$H(\boldsymbol{X}_{i+1,i+r_{n}}/u_{n})=1$ if and only if $i\leqslant
t_{j}{(N_{j})}$. Hence, ${\rm{SB}}_{j}(H)=t_{j}{(N_{j})}-(j-1)r_{n}$.

  * •

On $A_{j}$, ${\rm{DB}}_{j}(H)=r_{n}$.

Hence, on $A_{j-1}^{c}\cap A_{j}\cap A_{j+1}^{c}$, we have

$\displaystyle{\rm{SB}}_{j-1}(H)+{\rm{SB}}_{j}(H)-{\rm{DB}}_{j}(H)=t_{j}{(N_{j})}-t_{j}(1)\;.$

##### Explanation for Example 3.2.4.

  * •

On $A_{j-1}^{c}\cap A_{j}$: for $i\in I_{j-1}$, we have
$H(u_{n}^{-1}\boldsymbol{X}_{i+1,i+r_{n}})=1$ if and only if
$i+r_{n}-1\geqslant t_{j}(1)$. Hence,
${\rm{SB}}_{j-1}(H)=(j-1)r_{n}-(t_{j}(1)-r_{n})=jr_{n}-t_{j}(1)$.

  * •

On $A_{j+1}\cap A_{j+2}^{c}$: for $i\in I_{j+1}$, we have
$H(u_{n}^{-1}\boldsymbol{X}_{i+1,i+r_{n}})=1$ if and only if $i\leqslant
t_{j+1}{(N_{j+1})}$. Hence, ${\rm{SB}}_{j+1}(H)=t_{j+1}{(N_{j+1})}-jr_{n}$.

  * •

On $A_{j}\cap A_{j+1}$, for $i\in I_{j}$,
$H(u_{n}^{-1}\boldsymbol{X}_{i,i+r_{n}-1})\equiv 0$ if and only if
$i>t_{j}{(N_{j})}$ and $i+r_{n}-1<t_{j+1}(1)$. Thus,
${\rm{SB}}_{j}(H)=r_{n}-(t_{j+1}(1)-t_{j}{(N_{j})}-r_{n})_{+}$.

Thus, on $\\{A_{j-1}^{c}\cap A_{j}\cap A_{j+1}\cap A_{j+2}^{c}\\}$

$\displaystyle{\rm{SB}}_{j-1}+{\rm{SB}}_{j}+{\rm{SB}}_{j+1}-{\rm{DB}}_{j,j+1}=\left[\left(t_{j+1}{(N_{j+1})}-t_{j}(1)\right)-\left(t_{j+1}(1)-t_{j}{(N_{j})}-r_{n}\right)_{+}\right]\;.$

## Acknowledgement

The authors would like to thank Stanislav Volgushev for indicating the
expansion problem. The authors are grateful to Philippe Soulier for an
extensive discussion. Both authors were supported by an NSERC grant. A part of
this research was conducted during the second author’s stay at the Sydney
Mathematical Research Institute (October - November 2022). Rafał Kulik thanks
for the support and hospitality provided by SMRI.

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