ON A
REDUCTION
OFNON-COMMUTATIVE
REIDEMEISTER TORSION FORHOMOLOGY
CYLINDERSTAKAHIROKITAYAMA
1. INTRODUCTION
Let$\Sigma_{g,1}$ beacompactoriented surfaceofgenus$g\geq 1$ with
one
boundarycomponent. Homologycylinders
over
asurfacewere
first introducedbyGoussarov [4]and Habiro [6]intheir
surgery
theory of3-manifolds developedfor the study offinite-type invariants. In [3,9]Garoufalidis and Levine introduced the homology cobordism group of homolo$gy$ cylinders,
which
can
beseen
as an
enlargement of themappingclass groupof the surface. We denoteby$C_{g},{}_{1}C_{g,1}^{\iota rr}$ and $\prime H_{g,1}$ the monoid of homology cylinders
over
$\Sigma_{g,1}$, the submonoidconsisting of
irreducible
ones
as
3-manifold and the smooth homology cobordismgroup
respectively. TheJohnson filtrations $C_{g,1}[k],$ $\prime H_{g,1}[k]$ of$C_{g,1},$ $\prime H_{g,1}$
are
definedas
the kemels of the actionson
$\pi_{1}\Sigma_{g.1}/(\pi_{1}\Sigma_{g,1})_{k}$,where thelower central series $G_{k}$of
a
group $G$ is definedinductively by$G_{1}$ $:=$$G$and$G_{k+1}$ $:=[G_{k},G]$
.
Sakasai [15, 16] studied torsion invariants of homology cylinders with in general
non-commutative coefficientsand showed by the degrees of theseinvariants associated to elements
of$H^{1}(\Sigma_{g,1})$
as
a
reduction thatthe submonoids$C_{g,1}^{irr}\cap C_{g,1}[k]$ for $k\geq 2$ and$Ker(C_{g.1}arrow \mathcal{H}_{g.1})$
have abelianquotients isomorphicto$(Z_{\geq 0})^{\infty}$
.
Notethatsincetheconnectedsum
ofa
homologycylinder andahomology 3-sphereis anotherhomologycylinder,it is reasonabletorestrict
our
attention to$C_{g,1}^{j\gamma\gamma}$ in considering “size”of$C_{g,1}[k]$
.
Morita [12]showedbyusinghis “trace maps”definedin[11]that theabelianizationof$H_{g,1}[2]$hasinfiniterank.GodaandSakasai[5] showed
byusingsuturedFloer homology theory that$C_{g,1}^{irr}$ has
an
abelianquotientisomorphicto$(Z_{\geq 0})^{\infty}$.
Cha, Friedl and Kim [1] showed by using abelian torsion invariants that the abelianization of${}^{t}H_{g,1}$ contains
a
direct summand isomorphic to $(Z/2)^{\infty}$, and that the abelianization of$\prime H_{g.1}[2]$
contains
a
direct summandisomorphicto$(Z/2)^{\infty}$ andone
isomorphic to$Z^{\infty}$ if$g\geq 2$.
The aim of this note is to present another reduction ofnon-commutative torsion invariants
introducedin [8] and togiveanother approachtoSakasai’s result for$C_{g,1}^{irr}\cap C_{g.1}[k]$
.
Morepre-cisely,
we
considerthe coefficients ofthemaximumorder terms oftorsion invariants associatedtobi-ordersof$\pi_{1}\Sigma_{g,1}/(\pi_{1}\Sigma_{g,1})_{k}$and
use
themtoprove
that thegroup
completion of$C_{g.1}^{j\gamma\gamma}\cap C_{g.1}[k]$has
an
abeliangroup
quotientofinfiniterankfor$k\geq 2$.
In [8]we
can
findan
analogous workon
submonoids of$C_{g,1}^{\iota rr}$ associatedtosolvablequotients of$\pi_{1}\Sigma_{g,1}$.
Inthisnote all homology groupsandcohomologygroups
are
with respect to integralcoeffi-cients unless specifically noted.
2. HOMOLOGYCYLINDERS
First werecall thedefinitions ofhomology cylinders andtheir homology cobordisms. See
[7], [17] for
more
detailson
homology cylinders.Tosimplify notationwe often write$\Sigma,\pi$instead of$\Sigma_{g,1},\pi_{1}\Sigma_{g,1}$, respectively. We take a base
Definition
2.1.
A homology cylinder $(M, i_{\pm})$over
$\Sigma$ is defined to bea
compact oriented 3-manifold $M$together withembeddings$i_{+},$$i_{-};\Sigmaarrow\partial M$satisfyingthe following:(i) $i_{+}$ isorientation preserving and$i_{-}$ is orientation reversing,
(ii)$\partial M=i_{+}(\Sigma)\cup i_{-}(\Sigma)$ and$i_{+}(\Sigma)\cap i_{-}(\Sigma)=i_{+}(\partial M)=i_{-}(\partial M)$,
(iii) $i_{+}|_{\partial\Sigma}=i_{-}|_{\partial\Sigma}$,
(iv) $(i_{+})_{*},$$(i_{-})_{*}:$ $H.(\Sigma)arrow H_{*}(M)$
are
isomorphisms.Twohomologycylinders$(M, i_{\pm}),$ $(N, j_{\pm})$
are
calledisomorphicif thereexistsan
orientationpre-serving homeomorphism$f:Marrow N$ satisfying $j_{\pm}=foi_{\pm}$
.
We denoteby $C_{g,1}$ the setof allisomorphism classes of homology cylinders
over
$\Sigma_{g,1}$.
Aproduct operation
on
$C_{g,1}$ is givenbystacking:$(M, i_{\pm}) \cdot(N,j_{\pm});=(M\bigcup_{i_{-}\circ(j_{*})^{-1}}N, i_{+}, j_{-})$,
whichtums$C_{g.1}$ into
a
monoid. Theunitis givenbythestandard cylinder$(\Sigma\cross[0,1], id\cross l, u\cross O)$.
As pointedout in [5, Proposition 2. 4] there is
an
epimorphism $F;C_{g,1}arrow\theta^{3}$as
follows,where$\theta^{3}$ is
the monoid of homology 3-sphereswiththeconnected
sum
operation. For$(M, i_{\pm})\in$$C_{g.1}$,
we
can
write $M=M’\# M’’$, where $M’$ is the prime factor of $M$containing $\partial M$.
Then$F(M, i_{\pm})$ $:=M”$
.
Therefore it is reasonable to consider the submonoid $C_{g,1}^{irr}$ consisting of all homology cylinders whose underlying 3-manifoldsare
irreducible.Deflnition 2.2. Twohomology cylinders$(M, i_{\pm})\sim_{m}(N, j_{\pm})$
are
said to be homology cobordantif thereexists
a
compactorientedsmooth 4-manifoldsuch that:(i)$\partial W=M\bigcup_{i_{*}i_{-J_{-}^{arrow 1}}}\circ j_{+}^{-1},\circ(-N)$,
(ii) $H_{*}(M)arrow H_{*}(W),$$H_{*}(N)arrow H_{*}(W)$
are
isomorphisms.Wedenote by$\mathcal{H}_{g,1}$ the quotientsetof$C_{g.1}$ withrespect to theequivalence relation of homology
cobordism.
The monoid structure of $C_{g,1}$ naturally induces
a
group structure of$\prime H_{g.1}$.
The inverse of$[M, i_{\pm}]\in {}^{t}H_{g,1}$ is givenby $[-M,i_{\tau}]$
.
Weset$N_{k}$ $:=\pi/\pi_{k}$
.
For$(M, i_{\pm})\in C_{g,1},$$(i_{\pm})$.
:
$N_{k}arrow\pi_{1}M/(\pi_{1}M)_{i}$are
isomorphismsaccordingto Stallings’ theorem [18]. We define
a
homomorphism $\varphi_{k}:C_{g.1}arrow$ Aut$N_{k}$ by $\varphi_{k}(M, i_{\pm})$ $:=$$(i_{+})_{*}^{-1}\circ(i_{-}).$
.
By abuseof notationwe
also denoteby$\varphi_{k}$ the namrally induced homomorphism$\prime H_{g.1}arrow$ Aut$N_{k}$
.
Definition23. TheJohnson
filtrations
of$C_{g,1}$ and${}^{t}H_{g,1}$are
the sequences$...\subset C_{g,1}[k]\subset\cdots\subset C_{g,1}[2]\subset C_{g,1}[1]=C_{g,1}$,
$...\subset {}^{t}H_{g,1}[k]\subset\cdots\subset’H_{g,1}[2]\subset {}^{t}H_{g,1}[1]=lt_{g,1}$ respectively,where$C_{g.1}[k]$ and${}^{t}H_{g.1}[k]$
are
thekemels of$\varphi_{k}$.3. AREDUCTIONOF THETORSION HOMOMORPHISM
Nextwe review torsion invariantsofhomology cylinders and introduceanotherreductionof
the
group
where these torsion invariantsate defined,usinga
bi-order of the nilpotent quotient$N_{k}:=\pi/\pi_{k}$
.
Let$K$be
a
skew field. Wewrite write $K_{ab}^{x}$ for the abelianization of the unit group $K^{x}$.
Fora
finiteCW-pair$(X, Y)$ and ahomomorphism$\rho:Z[\pi_{1}X]arrow K$suchthatthe twisted homologygroup
$H_{*}^{\rho}(X, Y;K)$associatedto$\rho$vanishes,the Reidemeistertorsion$\tau_{\rho}(X, Y)\in K_{ab}^{x}/\pm\rho(\pi_{1}X)$We set$A_{k}$ $:=\pi_{k-1}/\pi_{k}$
.
Since $A_{k}$ is torsion-free for $aI1k,$$N_{k}$ hasa
finite filtrationofnormalsubgroups such that all successive quotient
are
torsion-free abeliangroups.
It is known thatfor such a
group
$G$ (called poly-torsion-free-abelian), $Q[G]$ isa
right (and left) Ore domain;namely$\mathbb{Q}[G]$ embedsin its classical right ringof quotients$\mathbb{Q}(G)$
$:=\mathbb{Q}[G](\mathbb{Q}[G]\backslash 0)^{-1}[13]$
.
For$(M, i_{\pm})\in C_{g,1}$,
we
denote by$\rho_{k}$thecompositionof the homomorphisms $Z[\pi_{1}M]arrow Z[\pi_{1}M/(\pi_{1}M)_{k}]arrow Z[N_{k}](i_{+})^{-1}arrow \mathbb{Q}(N_{k})$.
See[2,Proposition2. 10]fora
proofofthefollowing lemma.Lemma3.1. For$(M, i_{\pm})\in C_{g},{}_{\}}H_{*}^{\rho\iota}(M, i_{+}(\Sigma);\mathbb{Q}(N_{k}))=0$
.
Definition3.2. 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 followingpropositionis
a
version of[1,Proposition 3. 5] and [8,Corollary 3.10]. Seealso [16,Proposition6.6] for
a
relatedresult. The proof is almostsame as
that of[8,Corollaly3.10],and
so we
omitthe proof.Proposition
3.3.
Themap$\tau_{k}>\triangleleft\varphi_{k}:C_{g,1}arrow(\mathbb{Q}(N_{k})_{ab}^{\cross}/\pm N_{k})\triangleleft Aut(N_{k})$isa
homomorphism.Corollary
3.4.
Themap$\tau_{k}:C_{g,1}[k]arrow \mathbb{Q}(N_{k})_{ab}^{X}/\pm N_{k}$isa
homomorphism.We denote by$-:$ $Z[N_{k}]arrow Z[N_{k}]$the involution definedby$\overline{\gamma}=\gamma^{-1}$ for$\gamma\in N_{k}$ and naturally
extenditto$\mathbb{Q}(N_{k})$
.
We set$D_{k}:=\{\pm\gamma\cdot q\cdot\overline{q}\in \mathbb{Q}(N_{k})_{\ell rb}^{x};\gamma\in N_{k},q\in \mathbb{Q}(\Gamma_{m})_{ab}^{x}\}$
.
Thefollowing theorem isalso
a
versionof[1,Theorem3. 10] and[8,Corollary 3.13]. See themfor the proof.
Theorem3.5. Themap$\tau_{k}\cross\varphi_{k}:\mathcal{H}_{g,1}arrow(\mathbb{Q}(N_{k})_{ab}^{\cross}/D_{k})\triangleleft Aut(N_{k})$isahomomorphism.
Corollary3.6. Themap $\tau_{k}:^{t}H_{g,1}[k]arrow \mathbb{Q}(N_{k})_{ab}^{\cross}/D_{k}$ is
a
homomorphism.Abi-order $\leq$ of
a
group
$G$isa
totalorderof$G$satisfying thatif$x\leq y$, then$axb\leq ayb$ forall$a,b,$ $x,y\in G$
.
Agroup
$G$is calledbi-orderableif$G$admitsa
bi-order. It iswell-knownthatevery
finitely generatedtorsion-free nilpotent
group
is residually $p$for any prime$p.$ Rhemmlla [14] showedthata
groupwhichisresidually $p$for infinitelymany$p$is bi-orderable. Together with thefact that$N_{k}$ istorsion-ffee, wesee
that$N_{k}$isa
bi-orderable.Inthefollowing
we
fixa
bi-orderof$N_{k-1}$.
Wedefinea
map$c:Z[N_{k}]\backslash 0arrow \mathbb{Q}(A_{k})^{x}/\pm A_{k}$by$c( \sum_{\delta\in N_{k-1}}\sum_{\gamma\in N_{k},[\gamma]=\delta}a_{\gamma}\gamma)=[[\sum_{\gamma\in N_{k},[\gamma]\underline{-}\delta_{m}}a_{\gamma}\gamma)\gamma_{0}^{-1}]$,
where$\delta_{m\infty(}\in N_{k-1}$ is themaximumwithrespecttothe fixedbi-ordersuchthatfor
some
$\gamma\in N_{k}$
with $[\gamma]=\delta_{\max},$$a_{\gamma}\neq 0$,and$70\in N_{k}$ is
an
elementwith $[\gamma_{0}]=\delta_{mx}$.
The proof of the followinglemma is straightforward.
Lemma3.7. Themap$c;Z[N_{k}]\backslash 0arrow \mathbb{Q}(A_{k})^{X}/\pm A_{k}$ doesnotdepend
on
thechoice$of\gamma_{0}$ and isBythe lemma
we
havea
group
homomorphism$Q(N_{k})_{ab}^{\cross}/\pm N_{k}arrow \mathbb{Q}(A_{k})^{X}/\pm A_{k}$whichmaps
$f\cdot g^{-1}$ to $c(f)\cdot c(g)^{-1}$ for $f,g\in Z[N_{k}]\backslash 0$
.
By abuse of notation,we
use
thesame
letter $c$for the homomorphism. Since there is
a
namral section $Q(A_{k})^{\cross}/\pm A_{k}arrow \mathbb{Q}(N_{k})_{ab}^{\cross}/\pm N_{k}$ of$d$,$\mathbb{Q}(A_{k})^{\cross}/\pm A_{k}$
can
beseen as
adirectsummandof$\mathbb{Q}(N_{k})_{ab}^{X}/\pm N_{k}$.
Forirreducible$p,q\in Z[A_{k}]\backslash 0$,
we
write$p\sim q$ifthereexists$a\in A_{k}$such that$p=\pm a\cdot q$.
Since $Z[A_{k}]$isa
unique factorizationdomain,every$x\in \mathbb{Q}(A_{k})^{X}/\pm A_{k}$can
bewrittenas
$x= \prod_{[p]}[p]^{e}[p]$,where$e_{[p]}$ is
a
uniquely determined integer. Wehavean
isomorphism$e:Q(A_{k})^{\cross}/\pm A_{k}arrow\oplus_{[p]}Z$defined by$e(x)= \sum_{[p]}e_{[p]}$
.
Thuswe
obtaina
homomorphism$e\circ c\circ\tau_{k}:C_{g.1}[k]arrow\oplus_{[p]}Z=Z^{\infty}$.
4. CONSmUCnONANDCOMPUTrlON
Inthis section
we
systematicallyconstruct theimages of$e\circ c\circ\tau_{k}:C_{g.1}[k]arrow Z^{\infty}$.
For
nontrivia17
$\in\pi$anda
tame knot$K\subset S^{3}$,we
constructa
homology cylinder $M(\gamma, K)$as
follows.Let$*\in\Sigma$ be thebase point for$\pi$
.
We choosea
smoothpath$f:[0,1]arrow\Sigma$representing$\gamma$ such that$f^{-1}(*)=\{0,1\}$, anddefine$f:[0,1]arrow\Sigma\cross[0,1],$ $h:[0,1]arrow\Sigma\cross[0,1]$ by
$\tilde{f}(t)=$
$(f(t), t)$ and $h(t)=(*, 1-t)$
.
After pushed into the interior,$f\cdot h$ determinesa
tameknot $J\subset$$Int(\Sigma\cross[0,1])$
.
Let$E$,
be the complement ofan
open tubular neighborhood $Z$of $J$.
We takea
haming of$J$so
thata
meridian of$J$represents the conjugacy class ofthe generator of thekemel of$\pi_{1}\partial Zarrow H_{1}(\Sigma\cross[0,1])$ compatiblewith theorientationof$J$andthat
a
longitude of$J$ represents the conjugacyclass oftheimage of 7by$(i_{-})_{*}:\piarrow\pi_{1}E_{J}$.
Let$E_{K}$ betheexteriorof$K$
.
Now $M(\gamma, K)$ istheresult of$attaching-E_{K}$ to$E_{J}$ alongthe boundariesso
that alongimdeand
a
meridianof$K$correspondtoa
meridian anda
longimde of$J$respectively.Lemma4.1. Forallnontrivial$\gamma\in\pi$and allknots$K\subset S^{3},$$M( \gamma, K)\in C_{g.1}^{irr}\cap(\bigcap_{k}C_{g,1}[k])$
.
Proof.
If$K$isatrivialknot,then$M(\gamma, K)$ istheunit of$C_{g,1}$ for all nontrivial$\gamma\in\pi$,andthere isnothingtoprove.Inthefollowing
we
assume
that$K$is nontrivial.Since $E_{J}$ and $E_{K}$
are
both irreducibleand $\partial Z$ and$\partial E_{K}$are
both incompressible, $M(\gamma, K)$ isalsoirreducible.
Extending
a
degree 1 map $(E_{K}, \partial E_{K})arrow(Z,\partial Z)$ by the identity mapon
$E_{J}$,we
have$f:M(\gamma, K)arrow\Sigma\cross[0,1]$
.
The following commutative diagram ofisomorphisms shows that$M(\gamma, K)\in C_{g,1}[k]$ for all$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$
Proposition4.2. Let$\gamma\in\pi_{k}\backslash 1$
.
Then$\tau_{k+1}(M(\gamma, K))=[\Delta_{K}(\gamma)]$for
all$K$.
Proof.
Wehave the following shortexactsequencesoftwistedchain 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$,
where all thecoefficients
are
understoodtobe$\mathbb{Q}(N_{k})$.
Itis 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$
.
Thereforebythe homology longexact
sequences
$H_{*}^{\rho k}(E_{J}, i_{+}(\Sigma);\mathbb{Q}(N_{k}))=0$
.
Considering multiplicativityofReidemeister torsion inthe aboveexact 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
tocheck. Now these equations give the desired formula. $\square$ Recall that forevery
monoid $S$, there existsa
monoid homomolphism $g:Sarrow u(s)$ toa
group
$u(s)$ satisfyingthefollowing: Forevery
monoid homomorphism$f:Sarrow G$toa
group$G$,thereexistsauniquegrouphomomorphism$f’$: $\prime u(S)arrow G$such that$f=f’\circ g$
.
By theuni-versality$71(S)$ is uniquelydetermined uptoisomorphisms. Finally, usingthehomomorphism
$eoc\circ 7_{k}:C_{g,1}[k]arrow Z^{\infty}$,
we
give another proof of the following theorem which isa
direct corollaryofSakasai’s.Theorem4.3([16,Corollary6.16]). Thegroup$u(C_{g1}^{ir.r}\cap C_{g,1}[k])$ hasanabelian groupquotient
of infinite
rankfor
$k\geq 2$.Proof.
Let $\gamma\in\pi_{k-1}\backslash \pi_{k}$ and let $K\subset S^{3}$ bea
tame knot. By Lemma4.1 we see $M(\gamma, K)\in$$C_{g,1}^{\iota rr}\cap C_{g,1}[k]$
.
ByProposition4.2we
have$co\tau_{k}(M(\gamma, K))=[\Delta_{K}(\gamma)]$
.
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
a
knot$K\subset S^{3}$ such that $\Delta_{K}=p$, theimage of$eoco\tau_{k}:C_{g,1}^{irr}\cap C_{g,1}[k]arrow\oplus_{[p]}Z$ contains
a
submonoid isomorphicto$Z_{\geq 0}^{\infty}$.
Thereforetheimage oftheinducedmap$u(C_{g.1}^{irr}\cap C_{8,1}[k])arrow Z^{\infty}$is
a
free abeliangroupofinfiniterank,whichprovesthe theorem.a
Acknowledgment. The author wishes to express his gratimde to Tomotada Ohtsuki for his
encouragementandhelpful suggestions. The authorwould also liketothank theorganizersfor
inviting him to the stimulating workshop and all the participants for fmitful discussions and
advice. This research
was
supportedbyJSPS ResearchFellowshipsfor YoungScientists. REFERENCES[1] J.Cha,S. Friedl and T.Kim,The cobordismgroupofhomologycylinders,arXiv:0909.5580.
[2] T.D. Cochran,K. E.Orr,P. Teichner,Knotconcordance, Whitneytowersand$L^{2}$-signatures,Ann. of Math.
(2)157(2003)433-519.
[3] S. Garoufalidis and J. Levine, Tree-levelinvariantsofthree-manifolds, Massey products and the Johnson
homomorphism,Graphsandpattemsin mathematicsand theoretical physics, Proc.Sympos.Pure Math. 73,
[4] M.Goussarov,Finitetype invariants andn-equivalenceof3-manifolds,C. R. Acad. Sci. Paris Ser. I Math.
329(1999)517-522.
[5] H.Godaand T.Sakasai,Abelianquotientsofmonoids ofhomology$cylind\ell rs$, Geom. Dedicata151 (2011)
387-396.
[6] K.Habiro,Claspersandfnitetypeinvariantsoflinks,Geom. Topol.4(2000)1-83.
[7] K.Habiro and G. Massuyeau, From mappingclass groups tomonoidsofhomology cobordism: asurvey,
arXiv: 1003.2512.
[8] T. Kitayama,Homologycylindersofhigher-order,arXiv:1108.5903.
[9] J.Levine,Homologycylinders:anenlargementofthe mapping classgroup,Algebr. Geom. Topol.1 (2001)
243-270.
[10] J.Milnor,Whiteheadtorsion,Bull. Amer. Math. Soc. 72(1966)358426.
[11] S.Morita,Abelian quotients
of
subgroupsofthe mapping classgroupofsurfaces, DukeMath. J.70(1993)699-726.
[12] S.Morita,Symplectic automorphismgroupsofnilpotentquotients
offiundamental
groupsofsurfaces,Groupsof diffeomorphisms,Adv.Stud. Pure Math.52, Math.Soc. Japan, Tokyo(2008)443-468.
[13] D.Passman,The algebraicstructureofgrouprings,Wiley,New York(1977).
[14] A.H.Rhemtulla,Residually$F_{p}$-groups,formanyprimes p,areorderabk. Proc. Amer. Math. Soc.41(1973)
31-33.
[15] T.Sakasai,Higher-orderAlexander invariantsforhomologycobordismsofa surface,Intelligence of low
dimensionaltopology2006,Ser. KnotsEverythuing40, WorldSci.Publ., Hackensack,NJ(2007)271-278.
[16] T.Sakasai,$Th\ell$Magnus representation andhigher-orderAlexander invariantsfirhomologycobordismsof
$su\phi aces$,Algebr.Geom.Topol.8(2008)803-848.
[17] T.Sakasai,AsurveyofMagnusrepresentationsformapping class groups andhomologycobordismsof
sur-faces,arXiv:1005.5501,
[18] J.Stallings, Homology and central seriesofgroups,J. Algebra2(1965) 170-181.
[19] V.Turaev,Introductiontocombinatorialtorsions,Lectures inMathematics,ETHZ\"unich(2001).
RESEARCHINsminzroRMAimmc$s_{cmc\epsilon s,Kv\sigma roU_{NIV\Re srrY},KvoI\mathfrak{v}606-8502}$,JAPAN
