数理解析研究所講究録
第 1777 巻 2012 年 31-36

31

ON A REDUCTION OF NON-COMMUTATIVE REIDEMEISTER TORSION FORHOMOLOGY CYLINDERSTAKAHIRO KITAYAMA

1. INTRODUCTIONLet
be a compact oriented surface of genus $g\geq 1$ with one boundary component.
Homology cylinders over a surface were first introduced by Goussarov [4] and Habiro [6] intheir surgery theory of 3-manifolds developed for the study of finite-type invariants. In [3, 9]
Garoufalidis and Levine introduced the homology cobordism group of homolo$gy$ cylinders,which can be seen as an enlargement of the mapping class group of the surface. We denote byandthe monoid of homology cylinders over
, the submonoid consisting ofirreducible ones as 3-manifold and the smooth homology cobordism group respectively. The
Johnson filtrationsofare defined as the kemels of the actions on, where the lower central seriesof a group is defined inductively by$:=[G_{k},G]$ .
and
Sakasai [15, 16] studied torsion invariants of homology cylinders with in general noncommutative coefficients and showed by the degrees of these invariants associated to elementsofas a reduction that the submonoids
for $k\geq 2$ andhave abelian quotients isomorphic to. Note that since the connected sum of a homology
cylinder and a homology 3-sphere is another homology cylinder, it is reasonable to restrict ourattention toin considering “size” of. Morita [12] showed by using his “trace maps”[11]
defined inthat the abelianization of $H_{g,1}[2]$ has infinite rank. Goda and Sakasai [5] showedby using sutured Floer homology theory thathas an abelian quotient isomorphic to.
Cha, Friedl and Kim [1] showed by using abelian torsion invariants that the abelianization ofcontains a direct summand isomorphic to, and that the abelianization ofcontains a direct summand isomorphic toand one isomorphic to
if $g\geq 2$ .
The aim of this note is to present another reduction of non-commutative torsion invariants
introduced in [8] and to give another approach to Sakasai’s result for. More precisely, we consider the coefficients of the maximum order terms of torsion invariants associatedto bi-orders ofand use them to prove that the group completion ofhas an abelian group quotient of infinite rank for $k\geq 2$ . In [8] we can find an analogous workon submonoids ofassociated to solvable quotients of.
In this note all homology groups and cohomology groups are with respect to integral coefficients unless specifically noted.
$\Sigma_{g,1}$$C_{g},{}_{1}C_{g,1}^{\iota rr}$$\prime H_{g,1}$$\Sigma_{g,1}$$C_{g,1}[k],$$\prime H_{g,1}[k]$$C_{g,1},$$\prime H_{g,1}$$\pi_{1}\Sigma_{g.1}/(\pi_{1}\Sigma_{g,1})_{k}$

$G$

$G$

$G_{k}$$G_{1}$$:=$$G_{k+1}$$H^{1}(\Sigma_{g,1})$$C_{g,1}^{irr}\cap C_{g,1}[k]$$Ker(C_{g.1}arrow \mathcal{H}_{g.1})$$(Z_{\geq 0})^{\infty}$$C_{g,1}[k]$$C_{g,1}^{j\gamma\gamma}$$(Z_{\geq 0})^{\infty}$$C_{g,1}^{irr}$${}^{t}H_{g,1}$$(Z/2)^{\infty}$$\prime H_{g.1}[2]$$(Z/2)^{\infty}$$Z^{\infty}$$C_{g,1}^{irr}\cap C_{g.1}[k]$$\pi_{1}\Sigma_{g,1}/(\pi_{1}\Sigma_{g,1})_{k}$$C_{g.1}^{j\gamma\gamma}\cap C_{g.1}[k]$$\pi_{1}\Sigma_{g,1}$$C_{g,1}^{\iota rr}$

2. HOMOLOGY CYLINDERSFirst we recall the definitions of homology cylinders and their homology cobordisms. See[7], [17] for more details on homology cylinders.

To simplify notation we often write
point for in .
$\pi$$\partial\Sigma$$\Sigma,\pi$

instead of$\Sigma_{g,1},\pi_{1}\Sigma_{g,1}$, respectively. We take a base32
over is defined to be a compact oriented 3Definition 2.1. A homology cylinder$M$
satisfying the following:
manifold together with embeddings(i) is orientation preserving and is orientation reversing,$(M, i_{\pm})$$i_{+},$$i_{-}$$i_{+}$

(ii)
(iii)
(iv)

$\Sigma$$i_{-};\Sigmaarrow\partial M$

and$\partial M=i_{+}(\Sigma)\cup i_{-}(\Sigma)$$i_{+}|_{\partial\Sigma}=i_{-}|_{\partial\Sigma}$$(i_{+})_{*},$ $(i_{-})_{*}:$$i_{+}(\Sigma)\cap i_{-}(\Sigma)=i_{+}(\partial M)=i_{-}(\partial M)$,$H.(\Sigma)arrow H_{*}(M)$,are isomorphisms.

are called isomorphic if there exists an orientation preTwo homology cylinders
the set of allserving homeomorphism $f:Marrow N$ satisfying $j_{\pm}=foi_{\pm}$ . We denote by.
isomorphism classes of homology cylinders over$(M, i_{\pm}),$ $(N, j_{\pm})$$C_{g,1}$$\Sigma_{g,1}$

A product operation on$C_{g,1}$

is given by stacking:,$(M, i_{\pm}) \cdot(N,j_{\pm});=(M\bigcup_{i_{-}\circ(j_{*})^{-1}}N, i_{+}, j_{-})$.
which tums
into a monoid. The unit is given by the standard cylinder
follows,4]
as[5,
epimorphism
is
2.
there
an
Proposition
As pointed out in
where is the monoid of homology 3-spheres with the connected sum operation. For. Then, we can write $M=M’\# M’’$ , where $M’$ is the prime factor of $M$ containing$F(M, i_{\pm})$ $:=M”$ . Therefore it is reasonable to consider the submonoid
consisting of all
homology cylinders whose underlying 3-manifolds are irreducible.
$(\Sigma\cross[0,1], id\cross l, u\cross O)$$C_{g.1}$$F;C_{g,1}arrow\theta^{3}$$(M, i_{\pm})\in$$\theta^{3}$$\partial M$$C_{g.1}$$C_{g,1}^{irr}$

are said to be homology cobordant
Deflnition 2.2. Two homology cylinders
if there exists a compact oriented smooth 4-manifold such that:$(M, i_{\pm})\sim_{m}(N, j_{\pm})$(i),(ii) $H_{*}(M)arrow H_{*}(W),$ $H_{*}(N)arrow H_{*}(W)$ are isomorphisms.
$\partial W=M\bigcup_{i_{*}i_{-J_{-}^{arrow 1}}}\circ j_{+}^{-1},\circ(-N)$

We denote bycobordism.

$\mathcal{H}_{g,1}$

the quotient set of$C_{g.1}$

with respect to the equivalence relation of homology. The inverse ofnaturally induces a group structure of
The monoid structure ofis given by $[-M,i_{\tau}]$ .
:are isomorphisms according. For
We setAut byto Stallings’ theorem [18]. We define a homomorphism. By abuse of notation we also denote by the namrally induced homomorphism
Aut .
$\prime H_{g.1}$$C_{g,1}$$[M, i_{\pm}]\in {}^{t}H_{g,1}$$N_{k}$$:=\pi/\pi_{k}$$(M, i_{\pm})\in C_{g,1},$$(i_{\pm})$.

$N_{k}arrow\pi_{1}M/(\pi_{1}M)_{i}$$N_{k}$$\varphi_{k}:C_{g.1}arrow$$(i_{+})_{*}^{-1}\circ(i_{-}).$$\varphi_{k}(M, i_{\pm})$$:=$$\varphi_{k}$$N_{k}$$\prime H_{g.1}arrow$

Definition 23. The Johnson filtrations of$C_{g,1}$and${}^{t}H_{g,1}$

are the sequences$...\subset C_{g,1}[k]\subset\cdots\subset C_{g,1}[2]\subset C_{g,1}[1]=C_{g,1}$,$...\subset {}^{t}H_{g,1}[k]\subset\cdots\subset’H_{g,1}[2]\subset {}^{t}H_{g,1}[1]=lt_{g,1}$

respectively, where$C_{g.1}[k]$and${}^{t}H_{g.1}[k]$are the kemels of$\varphi_{k}$.

3. A REDUCTION OF THE TORSION HOMOMORPHISM
Next we review torsion invariants of homology cylinders and introduce another reduction of
the group where these torsion invariants ate defined, using a bi-order of the nilpotent quotient$N_{k}:=\pi/\pi_{k}$

.

. For
for the abelianization of the unit group
Let be a skew field. We write write$\rho:Z[\pi_{1}X]arrow
K$
$(X,
Y)$
homology
twisted
the
such
that
a finite CW-pairand a homomorphismgroup
associated to vanishes, the Reidemeister torsion
associated to is defined. See [10] and [19] for more details.
$K$$K_{ab}^{x}$$H_{*}^{\rho}(X, Y;K)$$\rho$$\rho$$K^{x}$$\tau_{\rho}(X, Y)\in K_{ab}^{x}/\pm\rho(\pi_{1}X)$33
We set. Sinceis torsion-free for $aI1k,$
has a finite filtration of normal
subgroups such that all successive quotient are torsion-free abelian groups. It is known thatfor such a group (called poly-torsion-free-abelian), $Q[G]$ is a right (and left) Ore domain;
namelyembeds in its classical right ring of quotients. For, we denote by
the composition of the homomorphisms$:=\pi_{k-1}/\pi_{k}$$A_{k}$$A_{k}$$N_{k}$$G$$\mathbb{Q}[G]$$\mathbb{Q}(G)$$(M, i_{\pm})\in C_{g,1}$$:=\mathbb{Q}[G](\mathbb{Q}[G]\backslash 0)^{-1}[13]$$\rho_{k}$.

$Z[\pi_{1}M]arrow Z[\pi_{1}M/(\pi_{1}M)_{k}]arrow Z[N_{k}](i_{+})^{-1}arrow \mathbb{Q}(N_{k})$See [2, Proposition 2. 10] for a proof of the following lemma.

Lemma 3.1. For$(M, i_{\pm})\in C_{g},{}_{\}}H_{*}^{\rho\iota}(M, i_{+}(\Sigma);\mathbb{Q}(N_{k}))=0$

Definition 3.2. We define a map.

$\tau_{k}:C_{g.1}arrow \mathbb{Q}(N_{k})_{ab}^{\cross}/\neq N_{k}$by.

$\tau_{k}(M, i_{\pm}):=\tau_{\rho_{k}}(M, i_{+}(\Sigma))$

The following proposition is a version of [1, Proposition 3. 5] and [8, Corollary 3.10]. See
also [16, Proposition 6. 6] for a related result. The proof is almost same as that of [8, Corollaly
3.10], and so we omit the proof.

Proposition 3.3. The map

Corollary 3.4. The map$\tau_{k}>\triangleleft\varphi_{k}:C_{g,1}arrow(\mathbb{Q}(N_{k})_{ab}^{\cross}/\pm N_{k})\triangleleft Aut(N_{k})$$\tau_{k}:C_{g,1}[k]arrow \mathbb{Q}(N_{k})_{ab}^{X}/\pm N_{k}$is a homomorphism.

is a homomorphism.

We denote by $Z[N_{k}]arrow Z[N_{k}]$ the involution defined byextend it to. We set$-:$for$\overline{\gamma}=\gamma^{-1}$$\gamma\in N_{k}$and naturally$\mathbb{Q}(N_{k})$$D_{k}:=\{\pm\gamma\cdot q\cdot\overline{q}\in \mathbb{Q}(N_{k})_{\ell rb}^{x};\gamma\in N_{k},q\in \mathbb{Q}(\Gamma_{m})_{ab}^{x}\}$.

The following theorem is also a version of [1, Theorem 3. 10] and [8, Corollary 3.13]. See themfor the proof.

Theorem 3.5. The map$\tau_{k}\cross\varphi_{k}:\mathcal{H}_{g,1}arrow(\mathbb{Q}(N_{k})_{ab}^{\cross}/D_{k})\triangleleft Aut(N_{k})$Corollary 3.6. The map$\tau_{k}:^{t}H_{g,1}[k]arrow \mathbb{Q}(N_{k})_{ab}^{\cross}/D_{k}$is a homomorphism.

is a homomorphism.

of a group is a total order of satisfying that if $x\leq y$ , then $axb\leq ayb$ for all$a,b,$ $x,y\in G$ . A groupis called bi-orderable if admits a bi-order. It is well-known that every
finitely generated torsion-free nilpotent group is residually for any prime Rhemmlla [14]showed that a group which is residually for infinitely many is bi-orderable. Together withthe fact thatis torsion-ffee, we see that is a bi-orderable.
In the following we fix a bi-order of. We define a mapbyA bi-order$G$$\leq$

$G$

$G$

$G$

$p$

$p$

$N_{k}$$p.$$p$$N_{k}$$N_{k-1}$$c:Z[N_{k}]\backslash 0arrow \mathbb{Q}(A_{k})^{x}/\pm A_{k}$$c( \sum_{\delta\in N_{k-1}}\sum_{\gamma\in N_{k},[\gamma]=\delta}a_{\gamma}\gamma)=[[\sum_{\gamma\in N_{k},[\gamma]\underline{-}\delta_{m}}a_{\gamma}\gamma)\gamma_{0}^{-1}]$

,

where
is the maximum with respect to the fixed bi-order such that for somewith, and $70\in N_{k}$ is an element with. The proof of the following
lemma is straightforward.
$\delta_{m\infty(}\in N_{k-1}$$\gamma\in N_{k}$$[\gamma]=\delta_{\max},$ $a_{\gamma}\neq 0$

Lemma 3.7. The map
a monoid homomorphism.

$c;Z[N_{k}]\backslash 0arrow \mathbb{Q}(A_{k})^{X}/\pm A_{k}$$[\gamma_{0}]=\delta_{mx}$does not depend on the choice$of\gamma_{0}$and is34
which maps
By the lemma we have a group homomorphism$c(f)\cdot c(g)^{-1}$ for $f,g\in Z[N_{k}]\backslash 0$ . By abuse of notation, we use the same lettertoof ,for the homomorphism. Since there is a namral section.
can be seen as a direct summand of. Sincesuch that, we write $p\sim q$ if there exists
For irreducible,
can be written as
is a unique factorization domain, every
is a uniquely determined integer. We have an isomorphism
where.
defined by $e(x)= \sum_{[p]}e_{[p]}$ . Thus we obtain a homomorphism$Q(N_{k})_{ab}^{\cross}/\pm N_{k}arrow \mathbb{Q}(A_{k})^{X}/\pm A_{k}$$c$$f\cdot g^{-1}$$Q(A_{k})^{\cross}/\pm A_{k}arrow \mathbb{Q}(N_{k})_{ab}^{\cross}/\pm N_{k}$$d$$\mathbb{Q}(N_{k})_{ab}^{X}/\pm N_{k}$$\mathbb{Q}(A_{k})^{\cross}/\pm A_{k}$$a\in A_{k}$$p,q\in Z[A_{k}]\backslash 0$$p=\pm a\cdot q$$x= \prod_{[p]}[p]^{e}[p]$$x\in \mathbb{Q}(A_{k})^{X}/\pm A_{k}$$Z[A_{k}]$$e:Q(A_{k})^{\cross}/\pm A_{k}arrow\oplus_{[p]}Z$$e_{[p]}$$e\circ c\circ\tau_{k}:C_{g.1}[k]arrow\oplus_{[p]}Z=Z^{\infty}$

4. CONSmUCnON AND COMPUTrlON.
In this section we systematically construct the images of$M(\gamma, K)$ as
cylinder,homology
we construct a
and a tame knot
For nontrivia17
be the base point for . We choose a smooth path $f:[0,1]arrow\Sigma$ representing
follows. Let
such that $f^{-1}(*)=\{0,1\}$ , and define $f:[0,1]arrow\Sigma\cross[0,1],$ $h:[0,1]arrow\Sigma\cross[0,1]$ bydetermines a tame knot$(f(t), t)$ and $h(t)=(*, 1-t)$ . After pushed into the interior,, be the complement of an open tubular neighborhood of . We take$Int(\Sigma\cross[0,1])$ . Leta haming of so that a meridian of represents the conjugacy class of the generator of thecompatible with the orientation of and that a longitude ofkemel of
. Let be the exterior of
represents the conjugacy class of the image of 7 by$M(\gamma,K)$so that alongimde$attaching-E_{K}$boundaries
along
the
to
of
is the result. Now
and a meridian of correspond to a meridian and a longimde of respectively.
$e\circ c\circ\tau_{k}:C_{g.1}[k]arrow Z^{\infty}$$K\subset S^{3}$$\in\pi$$*\in\Sigma$$\pi$$\tilde{f}(t)=$$\gamma$$J\subset$$f\cdot h$

$Z$

$E$

$J$

$J$

$J$

$J$

$J$

$\pi_{1}\partial Zarrow H_{1}(\Sigma\cross[0,1])$$(i_{-})_{*}:\piarrow\pi_{1}E_{J}$$K$$E_{K}$$E_{J}$$J$$K$

Lemma 4.1. For all nontrivial$\gamma\in\pi$

and all knots$K\subset S^{3},$ $M( \gamma, K)\in C_{g.1}^{irr}\cap(\bigcap_{k}C_{g,1}[k])$.

, and there isfor all nontrivial
is the unit ofis a trivial knot, thennothing to prove. In the following we assume that is nontrivial.
are both incompressible, $M(\gamma, K)$ is
Since
and
are both irreducible and andalso irreducible.
, we haveby the identity map on
Extending a degree 1 map$f:M(\gamma, K)arrow\Sigma\cross[0,1]$ . The following commutative diagram of isomorphisms shows that
$M(\gamma, K)\in C_{g,1}[k]$ for all:

Proof. If$M(\gamma, K)$$K$$\gamma\in\pi$$C_{g,1}$$K$$E_{J}$$\partial Z$$E_{K}$$\partial E_{K}$$E_{J}$$(E_{K}, \partial E_{K})arrow(Z,\partial Z)$$k$$\pi_{1}M(\gamma, K)/(\pi_{1}M(\gamma, K))_{k}$$\pi_{1}(\Sigma\cross[0,1])/(\pi_{1}(\Sigma\cross[0,1]))_{k}$$\square$Proposition 4.2. Let$\gamma\in\pi_{k}\backslash 1$. Then$\tau_{k+1}(M(\gamma, K))=[\Delta_{K}(\gamma)]$for all .
$K$

Proof. We have the following short exact sequences of twisted chain complexes:,$0arrow C_{*}^{\rho_{k}}(\partial E_{K})arrow C_{*}^{\rho_{k}}(E_{J}, i_{+}(\Sigma))\oplus C_{*}^{\rho\iota}(E_{K})arrow C_{*}^{\rho\iota}(M(\gamma, K), i_{+}(\Sigma))arrow 0$$0arrow C_{*}^{\rho_{i}}(\partial Z)arrow C_{*}^{\rho\iota}(E_{J}, i_{+}(\Sigma))\oplus C_{*}^{\rho\iota}(Z)arrow C_{*}^{\rho\iota}(\Sigma\cross[0,1], i_{+}(\Sigma))arrow 0$,

35where all the coefficients are understood to be$\mathbb{Q}(N_{k})$. It is easily checked that

$H_{*}^{\rho_{k}}(\partial E_{K};\mathbb{Q}(N_{k}))=H_{*}^{\beta k}(E_{K};\mathbb{Q}(N_{k}))=H_{*}^{\rho_{k}}(\partial Z;\mathbb{Q}(N_{k}))=H_{*}^{\rho_{k}}(Z;\mathbb{Q}(N_{k}))=0$.

Therefore by the homology long exact sequences$H_{*}^{\rho k}(E_{J}, i_{+}(\Sigma);\mathbb{Q}(N_{k}))=0$.

Considering multiplicativity of Reidemeister torsion in the above exact sequences we obtain,$\tau_{\beta k}(E_{J}, i_{+}(\Sigma))\cdot\tau_{\rho k}(E_{K})=\tau_{\rho_{k}}(\partial E_{K})\cdot\tau_{\rho\kappa}(M(\gamma, K), i_{+}(\Sigma))$$\tau_{\rho_{k}}(E_{J}, i_{+}(\Sigma))\cdot\tau_{\rho_{k}}(Z)=\tau_{\beta t}(\partial Z)\cdot\tau_{\rho_{k}}(M(id), i_{+}(\Sigma))$.

Here$\tau_{\rho_{k}}(E_{K})=[\Delta_{K}(\gamma)(\gamma-1)^{-1}]$$\tau_{\rho_{k}}(Z)=[(\gamma-1)^{-1}]$,,$\tau_{\beta k}(\partial E_{K})=\tau_{\rho_{k}}(\partial Z)=\tau_{\rho_{k}}(\Sigma\cross[0,1], i_{+}(\Sigma))=1$,

which are easy to check. Now these equations give the desired formula.

$\square$Recall that for every monoid , there exists a monoid homomolphism $g:Sarrow u(s)$ to agroup $u(s)$ satisfying the following: For every monoid homomorphism $f:Sarrow G$ to a group, there exists a unique group homomorphism :such that $f=f’\circ g$ . By the universality $71(S)$ is uniquely determined up to isomorphisms. Finally, using the homomorphism, we give another proof of the following theorem which is a direct
$S$

$G$

$f’$

$\prime u(S)arrow G$$eoc\circ 7_{k}:C_{g,1}[k]arrow Z^{\infty}$

corollary of Sakasai’s.

Theorem 4.3 ([16, Corollary 6.16]). The group$u(C_{g1}^{ir.r}\cap C_{g,1}[k])$of infinite rank for $k\geq 2$ .
Proof. Let$\gamma\in\pi_{k-1}\backslash \pi_{k}$$C_{g,1}^{\iota rr}\cap C_{g,1}[k]$and let$K\subset S^{3}$. By Proposition 4.2 we have

has an abelian group quotient

be a tame knot. By Lemma 4.1 we see $M(\gamma, K)\in$$co\tau_{k}(M(\gamma, K))=[\Delta_{K}(\gamma)]$.

Since it is well-known that for any $p\in Z[t, t^{-1}]$ with $p(t^{-1})=p(t)$ and $p(1)=1$ , there exists
such that $\Delta_{K}=p$ , the image of $eoco\tau_{k}:C_{g,1}^{irr}\cap C_{g,1}[k]arrow\oplus_{[p]}Z$ contains asubmonoid isomorphic to. Therefore the image of the induced map
is a free abelian group of infinite rank, which proves the theorem.
aa knot$K\subset S^{3}$$u(C_{g.1}^{irr}\cap C_{8,1}[k])arrow Z^{\infty}$$Z_{\geq 0}^{\infty}$

Acknowledgment. The author wishes to express his gratimde to Tomotada Ohtsuki for his
encouragement and helpful suggestions. The author would also like to thank the organizers for
inviting him to the stimulating workshop and all the participants for fmitful discussions andadvice. This research was supported by JSPS Research Fellowships for Young Scientists.
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$Th\ell$

RESEARCH INsminz roR MAimmc $s_{cmc\epsilon s,Kv\sigma roU_{NIV\Re srrY},KvoI\mathfrak{v}606-8502}$, JAPANE-mail address: kitayama@kurims.kyoto-u.ac.jp