Exponential growth of torsion in abelian coverings

42 Pages·2012·0.38 MB·English

by  Jean Raimbault

Algebraic & Geometric Topology12(2012)1331–1372 1331
Exponential growth of torsion in abelian coverings
JEAN RAIMBAULT
WeshowexponentialgrowthoftorsionnumbersforlinkswhosefirstnonzeroAlexan-
derpolynomialhaspositivelogarithmicMahlermeasure. Thisextendsatheoremof
SilverandWilliamstothecaseofanullfirstAlexanderpolynomialandprovidesa
partialsolutionforaconjectureoftheirs.
57M10;57M25,57Q10
Introduction
Let M beacompactthree–manifold;thehomologygroups H .M/ canbewrittenas
i
thedirectsums H .M/ ˚H .M/ ofafiniteabeliangroupwithafinite-rankfree
i tors i free
abeliangroup. Thetorsionsummandisnontrivialonlyfor i D1: H and H are Z
0 3
or 0,and theuniversalcoefficientstheoremforcohomology implies that H1.M;@M/
is free, and by Poincaré duality it follows that H .M/ is also torsion-free. On the
2
otherhand,thetorsionin H canbearbitrarilylarge(forexample,forlensspaces;see
1
belowforhyperbolicexamples)anditisbelievedthat“most”3–manifoldsshouldhave
aratherlargetorsion. ForexampleEKowalskishowsin[9,Proposition7.19]thatthe
first homology group of a “Dunfield–Thurston random 3–manifold” typically has a
large torsion subgroup. This paperis concernedwith the growth rateof theorder of
H .M / inasequenceoffinitecoverings M ofamanifold M. Theleastprecise
1 N tors N
question that can be asked is whether it is exponential in the degree or not, that is,
whetherthesequence logjH1.MN/torsj hasapositivelimit(orlimitsuperior). Thisshall
Œ(cid:25)1.M/W(cid:25)1.MN/(cid:141)
bepartiallyansweredhereinthecasewherethe M areabeliancoveringsconverging
N
to a free abelian covering of M. The main motivation to study this question was to
provideapartialresulttowardsaconjectureofSilverandWilliamsonthegrowthrateof
torsionnumbersofabeliancoveringsofcomplementsoflinks(seeSilver–Williams[24,
Conjecture6.1]or(0-1)below).
Historically,thefirstcontextwherethegrowthoftorsioninthehomologyofcoverings
hasbeenstudiedisthatofcycliccoveringsofaknotcomplement. Let K beanopen
knottedsolidtorusinthethree-sphereand M DS3(cid:0)K;then M isacompactthree-
manifoldwith H1.M/DZ. Thuswecanconsidertheinfinitecycliccovering M(cid:128) of
M anditsfinitequotients M ,whicharethefinitecoveringsof M correspondingto
N
Published: 23June2012 DOI:10.2140/agt.2012.12.1331
1332 JeanRaimbault
the maps (cid:25) .M/!Z!Z=NZ. If (cid:129) is the Alexander polynomial of K (the first
1
Alexanderpolynomialofthe ZŒZ(cid:141)–module H1.M(cid:128)/,see1.2)thentheso-calledFox
formulasaysthatforall N suchthat H .M / hasrankonewehave
1 N
Y
jH .M / jD j(cid:129).(cid:16)/j:
1 N tors
(cid:16)ND1
Itisknownthat (cid:129) isalways nonzero. Adiophantineinequalitydue toGelfondthen
allows toshowthat when N tendsto infinity 1 P logj(cid:129).(cid:16)/j convergesto the
N (cid:16)ND1
logarithmicMahlermeasureof (cid:129),thatis,
Z
m.(cid:129)/D logj(cid:129).z/jdz
T1
(here T1 is theunit circlein C endowedwith theLebesgue probabilitymeasure); see
Lemma2.6beloworSilver–Williams[25,Proposition2.8]. Let d betheproductofall
integers l such that the lth cyclotomic polynomial divides (cid:129): it is known (see Silver–
Williams [25, Proposition 2.2] or Gordon [6, Theorem 4.1(ii)]) that H .M ;Q/Š
1 N
H1.MNCd;Q/. Wegetthat
lim (cid:0)1 logjH .M / j(cid:1)Dm.(cid:129)/I
N!1 N 1 N tors
djN(cid:0)1
in particular, if m.(cid:129)/ > 0 an infinite subsequence of the torsion numbers tends to
infinity with exponential growth. This result answered a question of Gordon (who
provedin[6]thatwhen m.(cid:129)/D0 thetorsionnumbersareperiodicin N)andhasfirst
beenprovedbyGonzález-AcuñaandShortin[5]andindependentlybyRobertRiley
in[20]. Sincethereexistshyperbolic knotswhoseAlexanderpolynomialhaspositive
Mahlermeasure(forexamplethefigure-eight)thisprovesinparticularthatthereexists
(noncompactwithfinitevolume)hyperbolic manifolds witharbitrarilylargetorsionin
their H .
1
Thesimilarsettingforacollectionof m>1 linkedsolidtori L inthethree-sphereisto
considerthecoverings M of M DS3(cid:0)L givenbythemaps (cid:25) .M/!H .M/Š
H 1 1
Zm ! Zm=H where H is a finite-index subgroup. We want to study the torsion
numbersasthequantity
˛.H/Dminfmax jv jW06DvD.v ;:::;v /2Hg
i i 1 m
tends to infinity. Fox’s formula admits a generalization due to Mayberry and Mura-
sugi[18];seealsoPorti[19]foraproofusingReidemeistertorsion. However,because
theBettinumbersmayverywelltendtoinfinityas ˛.H/ does,itmaynotbeapplicable
toaninfinitesequenceofcoverings. Thisforbidsanynaïveapplicationofthepreceding
Algebraic & Geometric Topology,Volume12(2012)
Exponentialgrowthoftorsioninabeliancoverings 1333
schemeofprooftothiscase. Moreover,twootherobstructionsarisewhenconsidering
links: the Alexander polynomial may be zero, and the generalisation of Gelfond’s
estimateneededtoprovetheconvergenceofRiemannsumstotheMahlermeasureis
notknowntoholdforpolynomialsinseveralvariables.
ByusingmethodsfromalgebraicdynamicalsystemsDanielSilverandSusanWilliams,
in[24],wereabletoshowthatifthefirstAlexanderpolynomial (cid:129).L/ of L isnonzero
then
logjH .M / j Z
limsup 1 H tors Dm.(cid:129).L//WD logj(cid:129).L/j
ŒG WH(cid:141)
˛.H/!1 Tm
andthatthisholdswithalimitinthecaseofaknot(thisisalsoprovedin[25]). They
alsoconjecturethat,inthegeneralcase,thefollowinglimitshouldholdwhere (cid:129) is
i
thefirstnonzeropolynomialinthesequenceofAlexanderpolynomialsof L:
logjH .M / j
(0-1) lim 1 H tors Dm.(cid:129) .L//
˛.H/!1 ŒG WH(cid:141) i
TheMahlermeasureoftheAlexanderpolynomialofaknotcanbeinterpretedasthe
`2–torsionoftheinfinitecycliccovering M(cid:128) ofitsexterior M (seeforexample(1-5)
below). In[1]NicolasBergeronandAkshayVenkateshusedthistogiveanewproof
ofthetheoremofSilverandWilliamsinthecaseofaknot: themainpointisthatthe
nonnullityof theAlexanderpolynomialguarantees that M(cid:128) is `2–acyclic, andinthis
casethe approximationof `2–torsionby Reidemeistertorsionsis known. They then
show that the growth of torsion numbers is the same as that of Reidemeister torsion
andobtain ageneralization ofthetheorem forknots (see[1, Theorem7.3]). Forlinks
withnonzerofirstAlexanderpolynomialtheirproofcanbeadapted;howevertheresult
obtainedisslightlyweakerthanSilverandWilliams’.
The aimof thiswork isto generalisetheir resultsto thecase ofa nullfirst Alexander
polynomial,thatis,thatofanon–`2–acyclicinfinitecovering. Thisyieldsthefollowing
result,whichfollowsfromthemoregeneralTheorem3.1.
Theorem0.1 Withnotationasin(0-1)wehave
logjH .M / j
limsup 1 H tors (cid:21)m.(cid:129) .L//:
ŒG WH(cid:141) i
˛.H/!1
The same scheme of proof can be applied to homology with coefficients in a local
system. Let V be a free Z–module of finite rank and (cid:31) a representation of (cid:25) .M/
1
to GL.V/. Thenwecandefinehomologytwistedby (cid:31) forallcoveringsof M,also
called homology with coefficients in the local system defined by (cid:31). See Hatcher [7,
Algebraic & Geometric Topology,Volume12(2012)
1334 JeanRaimbault
Chapter3.H]fordefinitions. Let (cid:129) .L;(cid:31)/ bethefirstnonzeroAlexanderpolynomialof
i
H1.M(cid:128)IV/ (theAlexanderpolynomialswithnontrivialcoefficientsareusuallycalled
twistedAlexanderpolynomials). Weget(seealsoSilver–Williams[26,Theorem3.10])
logjH .M IV/ j
limsup 1 H tors (cid:21)m.(cid:129) .L;(cid:31)//:
ŒG WH(cid:141) i
˛.H/!1
In the case ofa sequence of cycliccoverings converging to apossibly non–`2–acyclic
infinitecycliccoveringtheproofsyieldamorepreciseresult,generalising[25,Theo-
rem2.10].
Theorem 0.2 If M is a compact n–manifold (with boundary) with a surjection
(cid:25) .M/!Z let M be the cyclic covering of M associated to (cid:25) .M/!Z=NZ
1 N 1
and M(cid:128) theinfinitecycliccovering. Thenwehaveforall i D1;:::;n(cid:0)1:
logjH .M / j
lim i N tors Dm.(cid:129) /
i
N!1 N
where (cid:129)i isthefirstnonzeroAlexanderpolynomialof Hi.M(cid:128)/.
ThangLehasrecentlyprovedtheequalitylackinginTheorem0.1inthepreprint[13].
His proof uses methods slightly different from ours, namely a Bourbaki theorem on
“pseudo-zeromodules”andthedynamicalversionofapproximation(duetoLind;see
Schmidt[22,Theorem21.1]). Healsogivesadifferentconstructionoftheapproximat-
ingsequencesconstructedinProposition2.7below.
Finally, notethat throughoutthis paperwework withthree-manifolds withboundary,
thatis,weconsiderregularcoveringsof S3(cid:0)L insteadofthecorrespondingbranched
(overcorecircles of L)coveringsof S3,which areclosedthree-manifolds. However,
thegrowthof torsion homologyin abelian coverings isthesame forthetwosequences
ofmanifolds,asshowninLe[13,4.4](forknotsthetwohomologiesdifferonlybya
infinitecyclicdirectfactor).
Warningsandoutline
Most of the techniques and results used here are fairly elementary with one notable
exception: I have found more practical and efficient to use the Fuglede–Kadison
determinantasdefinedinthebook[17]ofWolfgangLück,andallitspropertiesproved
thereininsteadoftryingtodoeverythingsolelyintermsofMahlermeasures(inany
case,itistherighttooltousetodefine `2–torsion). However,thereaderneednotbe
familiar with the Fuglede–Kadison determinant since all those of its properties that
weusearerecalled. Onewarningaboutnotations: Ihavechosentodenoteby M the
Algebraic & Geometric Topology,Volume12(2012)
Exponentialgrowthoftorsioninabeliancoverings 1335
multiplicativeMahlermeasuregivenby exp.m/ sinceIalreadyusethecasual M to
denotemodulesandmanifolds.
The paper is organised as follows: the first section reviews various classical and
`2–invariants for ZŒG(cid:141)–modules. In particular, I give there the computation of the
`2–torsion in terms of Mahler measures of Alexander polynomials. In the second
sectionIgiveaproofoftheapproximationoftheFuglede–Kadisondeterminantinthe
free Abelian case since there was (to the best of my knowledge) no complete proof
of this result available in the literature. In the third section we prove Theorems 0.1
and0.2. Thefourthsectioncontainssomefurtherremarksaboutthree-manifolds. The
appendicescontainproofsthatIjudgedtoocumbersomeforafirstreading;Ihopethat
theirbeingrelegatedtherewillmakethemainthrustofthepapermoreapparentand
intelligible.
Acknowledgements
ThesubjecttreatedherehasbeenproposedtomebymyPhDadvisor,NicolasBergeron.
Igratefullythankhimforthisaswellasforthenumeroussuggestionshemaderegarding
theredactionofthispaperandthemathinside. IalsowishtothankPierreVogelwho
tried to explain to me the Alexander polynomial. I have benefited from conversations
withThangLeregardinghispaper[13]andrelatedtopics,aswellashispointingout
a deadly mistake in a previous proof of Theorem 3.7. Also, the Bombieri–Zannier
theorem used in Appendix A has been brought to my attention by the notes from a
lecture he gave. Finally, I thank Takahiro Kitayama for reading carefully a previous
versionofthispaperandtherefereeforspottingmanytyposandminorflaws.
1 Review of classical and `2–invariants
Inthissection G shalldenoteafreeAbeliangroupofrank m;weidentifythegroup
ring ZŒG(cid:141) withtheLaurentpolynomials ZŒt˙1;:::;t˙1(cid:141) throughthechoiceofafree
1 m
generating set t ;:::;t for G. In this section, we review various invariants for
1 m
G–modulesand G–complexes.
1.1 Rank,vonNeumanndimensionand `2–Bettinumbers
RecallthatafinitelygeneratedHilbert N.G/–moduleisaHilbertspacewithaunitary
actionof G,whichadmitsa G–equivariantsurjectionof `2.G/n forsomeinteger n
(weshall notuseexplicitlythe vonNeumann algebra N.G/ inthe sequel,sowe shall
simplycallthesefinitelygenerated N.G/–modules). Givenasubmodule M (cid:26)ZŒG(cid:141)n,
Algebraic & Geometric Topology,Volume12(2012)
1336 JeanRaimbault
theclosure of M ˝C isa N.G/–submoduleof `2.G/n, inparticular itis afinitely
generated N.G/–module. ThevonNeumanndimensionofa N.G/–moduleisdefined
forany discretegroup(see Lück[17,Definition 1.10]): it assignstoa nonzerofinitely
generated N.G/–moduleapositiverealnumberandsatisfiesthesamebasicproperties
thatthedimensionofavectorspacedoes(see[17,Theorem1.12]). Ingeneralitisnot
easytocompute. However,inourcasethegroupring ZŒG(cid:141) hasafractionfield Q.G/
identified with the field of rational functions in m variables Q.t ;:::;t /. We can
1 m
thendefinetherankof M asthedimensionofthe Q.G/–vectorspace M ˝Q.G/.
Lemma1.1 Let M beasubmoduleof CŒG(cid:141)n;thenthevonNeumanndimensionof
itsclosurein `2.G/n isequaltoitsrank.
Proof Thisiseasy todeducefromthetwo lemmasbelow: according toLemma1.2
we can pick a free module L(cid:26)M such that M=L is torsion; then the rank of M
isequaltothatof L,andbyLemma1.3thecompletionof L isequaltothatof M.
Thusit suffices toshow thelemmafora freesubmodule L. Let r bethe rankof L;
then L istheimageofaninjectivemorphismfrom CŒG(cid:141)r to CŒG(cid:141)n,whichextendsto
anisomorphismof N.G/–modulesfrom `2.G/r totheclosureof L,sothatitsvon
Neumanndimensionis r.
Lemma 1.2 Let M bea finitelygenerated ZŒG(cid:141)–module. If L(cid:26)M isa maximal
rankfreesubmodule(whichalwaysexists)thenthequotient M=L istorsion.
Proof If a ;:::;a is a basis for the finite-dimensional vector space M ˝C.G/
1 r
such that a 2M for all i, then themodule LDP ZŒG(cid:141)a is amaximal rank free
i i i
submodule. If f isthelowestcommonmultipleofthedenominatorsofthecoordinates
of a generating family for M in the basis .a ;:::;a / we have M (cid:26)fL; thus f
1 r
annihilates M=L.
Lemma1.3 Thequotientofa CŒG(cid:141)–module M byasubmodule M0 istorsionifand
0
only M isdensein M.
Proof Ifthequotientistorsionthenforany x2M theintersectionof CŒG(cid:141)x with
0
M isequalto Ix forsomenonzeroideal I;thusitsufficestoprovethatanidealis
densein CŒG(cid:141). Thisistrue: anyideal I containsanonzeroprincipalideal .f/ (since
weareinaNoetherianUFD)andtheoperatorof `2.G/ inducedbymultiplicationby
f isinjective,soithasdenseimageby[17,Lemma1.14].
If M0 isdense thenits `2 completionequals thatof M andso itsrank, which equals
itsVonNeumanndimension,isthesameasthatof M. Nowitisclearthatthequotient
of a module by a submodule of maximal rank is torsion (see the proof of the above
lemma).
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Exponentialgrowthoftorsioninabeliancoverings 1337
Inthesameway,givenacomplex C(cid:3);d(cid:3) offreefinitelygenerated ZŒG(cid:141)–moduleswe
candefine itsreduced `2–homology: the C ˝C canbe endowed witha G–invariant
i
prehilbertianinnerproduct(wherewepicka ZŒG(cid:141)–basisanditsimagesunder G tobe
.2/
anorthonormalbasis)andthedifferentials d extendtoboundedoperators d from
i i
Ci.2/ to Ci.(cid:0)2/1. Thereduced ith `2–homologymodule Hi.2/.C/ of C(cid:3) isthendefined
tobetheN.G/–moduleker(cid:0)d.2/(cid:1)=im(cid:0)d.2/ (cid:1)andtheith`2–Bettinumberb.2/.C/to
i iC1 i
beitsdimension. ItfollowsfromtheLemmaabovethatwehave b.2/.C/Drk.H .C//
i i
(seealso[17,Lemma1.34]). Thecomplex C(cid:3) isthus `2–acyclicifandonlyall Hi.C/
are ZŒG(cid:141)–torsionmodules.
1.2 AlexanderpolynomialsandFuglede–Kadisondeterminant
Wenowpresenttheanaloguefor ZŒG(cid:141)–modulesoftheorderofthetorsionsubgroup
for Z–modules,andtheFuglede–KadisondeterminantwhichislinkedtotheMahler
measure.
Alexander polynomials of ZŒG(cid:141)–modules Let M be a finitely generated ZŒG(cid:141)–
moduleandA2Mm;n.ZŒG(cid:141)/apresentationmatrixforM,thatis,MDZŒG(cid:141)m=AZŒG(cid:141)n.
Wedefinethe lthelementaryidealof M astheidealof ZŒG(cid:141) generatedbythe .m(cid:0)l/–
minors of A. This definition makes sense since a given presentation matrix for M
can be made into any other one by a sequence of elementary transformations that
do not affect the elementary ideals (see Turaev [27, Lemma 4.4]). Since ZŒG(cid:141) is a
Noetherian UFD, we can take the greatest common divisor of an ideal; we thus put
(cid:129) .M/Dgcd.I /,andcallitthe lthAlexanderpolynomialofthemodule M. The
l l
Laurentpolynomial (cid:129) .M/ isofcoursedefinedonlyuptomultiplicationbyaunitof
l
ZŒG(cid:141) anditiszeroforall l (cid:20)rk.M/. Itdependsonlyonthe ZŒG(cid:141)–torsionsubmodule
of M;infactwehave(see[27,Chapter4])
(cid:129)rk.M/Cl.M/D(cid:129)l.Mtors/:
Inparticular,itfollowsfromthisequalitythat (cid:129) .M/D1 foratorsion-freemodule
r
M of rank r. We remark that (cid:129) is a multiplicative invariant, that is, if we have
0
a short exact sequence 0 ! M0 ! M ! M00 ! 0 of modules, then (cid:129) .M/ D
0
(cid:129) .M0/(cid:129) .M00/. Toprovethiswecansupposethemodulesaretorsion(ifnotboth
0 0
sidesoftheequalityarezero). Thus M0;M00 havepresentationmatrices A0;A00 which
areofmaximalranks m0;m00;then M hasapresentationmatrixoftheform
(cid:18)A0 (cid:3) (cid:19)
AD ;
00
0 A
whose nonzero .m0Cm00/–minors are products of m0–minors of A0 with m00–minors
00
of A .
Algebraic & Geometric Topology,Volume12(2012)
1338 JeanRaimbault
Notethatthe Mahlermeasureofthe Alexanderpolynomial (cid:129) .M/ isa well-defined
l
positiverealnumbersincetheunitsof ZŒG(cid:141) arethemonomials ˙tv for v2Zm,whose
Mahlermeasureisequaltoone.
Fuglede–Kadisondeterminant TheFuglede–Kadisondeterminantisdefinedforany
N.G/–modulemorphism,see Lück[17,Definition3.11]; itisaninfinite-dimensional
0
analogue of the det introduced in the next subsection. Here we only need it for
thosemorphismsthatareinducedfrommorphismsbetweenfree ZŒG(cid:141)–modules. Itis
alwaysnonzerowhenthemodulesarefinitelygenerated. If A isa .n(cid:2)n0/–matrixwith
coefficientsin ZŒG(cid:141) weabusivelydenoteby A thecontinuousoperatorfrom `2.G/n0
to `2.G/n thatitinduces,andby detN.G/.A/ itsFuglede–Kadisondeterminant. We
listsomeofitspropertiesbelowforthereader’sconvenience:
(i) If A issquarewithnonzerodeterminant,thenwehave:
(1-1) detN.G/.A/DM.detA/:
f g
(ii) [17,Theorem3.14(1)]If U !V !W , f hasdenseimageand g isinjective
then
detN.G/.gıf/DdetN.G/.f/detN.G/.g/:
(iii) [17,Theorem3.14(2)]If f hasdenseimageand f isinjectivethen
1 2
(cid:18)f f (cid:19)
detN.G/ 01 f3 DdetN.G/.f1/detN.G/.f2/:
2
(iv) [17,Lemma3.15(3)]Wehave
detN.G/(cid:16)fˇˇˇikmer..ff//?(cid:17)DdetN.G/.f/:
(v) [17,Lemma3.15(4)]Let f(cid:3) betheadjointof f ,then
q
detN.G/.f(cid:3)/DdetN.G/.f/D detN.G/.ff(cid:3)/:
Weprove(i),whichisleftasanexercisein[17]. Let ADLPU beaBruhatdecomposi-
tionof A overthefield C.G/,thatis, P isapermutationmatrix, L isalowertriangular
matrix with diagonal coefficients all equal to 1 and U is an upper triangular matrix
withdiagonalcoefficients u ;:::;u 2C.G/. Let q bethelcmofthedenominators
1 n
of the coefficients of L and U. We get detN.G/.q2A/ D detN.G/.qLPqU/; The
permutation matrix P is diagonalisable over C with modulus one eigenvalues, and
thus detN.G/.P/D1. Fromproperties(ii)and(iii)abovefollowfirst
detN.G/.q2Id/detN.G/.A/DdetN.G/.qL/detN.G/.P/detN.G/.qU/;
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Exponentialgrowthoftorsioninabeliancoverings 1339
andthen
detN.G/.A/DM.q/(cid:0)2nM.qn/M.qnu1:::un/
DM.u :::u /
1 n
DM.det.A//:
1.3 Reidemeistertorsionand `2–torsion
Reidemeister torsion Let us first recall some elementary definitions: if f is a
map between two finite-dimensional hermitian spaces V;W we define det0.f/ D
det.f(cid:3)fjker.f/?/ (the product of all nonzero eigenvalues of the self-adjoint map
f(cid:3)f ). Alatticeof V isdefinedtobea Z–submodule L ofrank dim.V/ spanning V
over C;itsvolumeisdefinedtobetheabsolutevalueofthedeterminantofanymap
0
sendinganorthonormalbasisof V toa Z–basisof L. If V isasubspaceof V such
that V0\L spans V0,thenthisisalsotrueforitsorthogonalcomplement V00 andwe
have
(1-2) vol.L/Dvol.L\V0/vol.L00/Dvol.L0/vol.L\V00/
where L0;L00 aretheorthogonalprojectionsof L on V0;V00. From(1-2)itiseasyto
deducethe“metricrankformula”
det0.f/vol.L/
(1-3) vol.im.f//D :
vol.kerf/
Now let C(cid:3);d(cid:3) be a finite complex of free ZŒG(cid:141)–modules; we pick for each i a
Z–basis of Ci and endow C(cid:3)˝C with the inner product where all these bases are
orthonormal. Thefreepart H .C/ ofthehomologyisalatticein H .C˝C/,which
i free i
itselfcanbeseenastheorthogonalof im.diC1/ in ker.di/˝C. ThentheReidemeister
torsiondefinedby
(cid:28).C(cid:3)/DYdet0.di/.(cid:0)1/i
i
doesnotdependonthechoiceofthebasesandisinfactgivenbytheequality
(1-4) (cid:28).C(cid:3)/DY(cid:18) jHi.C/torsj .(cid:0)1/iC1(cid:19):
vol.H .C/ /
i free
i
Theproofofthisequalityisasimpleexerciseusing(1-3).
Algebraic & Geometric Topology,Volume12(2012)
1340 JeanRaimbault
`2–torsion Let C(cid:3);d(cid:3) beafinitecomplexoffree ZŒG(cid:141)–modules. Byanalogywith
thefinite-dimensionalcaseweput
(cid:28).2/.C/DYdetN.G/.di/.(cid:0)1/i:
i
Inourparticularcasethiscanbecomputed:
Proposition1.4 Inthesituationabove,supposethat C(cid:3) is `2–acyclic. Thenwehave
(1-5) (cid:28).2/.C/DYM(cid:0)(cid:129) .H .C//(cid:1)
0 i
i
Proof ThiscanbededucedwithouttoomucheffortfromtheequalityinTuraev[27,
Theorem4.7]. Wedo notdosohere, andinsteadprovearesult whichisalsovalidfor
non `2–acycliccomplexes(LemmaB.3),followingthelinesoftheproofof(1-4). The
equality(1-5)followsimmediatelyfromthisresult.
Example: link complements Let X be a CW–structure on the complement of a
regularneighbourhood L ofalinkinthethree-sphere,andput G DH1.X/. Let X(cid:128)
be the maximal abelian cover of X; then X(cid:128) is a G–CW–complex. The only Hi.X(cid:128)/
thatcancontain ZŒG(cid:141)–torsionis H ,anditisknownthatinthecaseofaknot K, H
1 1
isinfactatorsionmodule. InthiscasewegetfromProposition1.4that
(cid:28).2/.X(cid:128)/DM.(cid:129).K//
where (cid:129).K/ istheAlexanderpolynomialof K. ForalinkwithnonzerofirstAlexander
polynomial (cid:129).L/ we get in the same way that (cid:28).2/.X(cid:128)/DM.(cid:129).K//; if (cid:129).L/D0
thenthe `2–torsiondependsontheCW–structure;theprecisedependanceiscomputed
inLemmaB.3.
2 Approximation for the Fuglede–Kadison determinant
Let H (cid:26)G beasubgroupand M a ZŒG(cid:141)–module. Wedenoteby M the ZŒG=H(cid:141)–
H
moduleobtainedbyrestrictingfrom G to G=H:
MH DZŒG=H(cid:141)˝ZŒG(cid:141)M
Thereisa G=H–equivariantisomorphismfrom M tothequotientmodule M=.H (cid:0)
H
1/M. If fW ZŒG(cid:141)n !ZŒG(cid:141)n0 is a linear map of free modules, f is the map from
H
ZŒG=H(cid:141)n!ZŒG=H(cid:141)n0 thatitinduces. Therestricteddeterminant det0.f / isalways
H
Algebraic & Geometric Topology,Volume12(2012)