Table Of ContentTHE TWISTED ALEXANDER POLYNOMIAL FOR FINITE
ABELIAN COVERS OVER THREE MANIFOLDS WITH
BOUNDARY
JE´ROˆMEDUBOISANDYOSHIKAZUYAMAGUCHI
2 Abstract. We provide the twisted Alexander polynomials of finite abelian
1 covers over three–dimensional manifolds whose boundary is a finite union of
0 tori. Thisisageneralizationofawell–knownformulafortheusualAlexander
2 polynomialofknotsinfinitecyclicbranchedcoversoverthethree–dimensional
sphere.
n
a
J
6
1
1. Introduction
] The classical Alexander polynomial is defined for null–homologous knots in ra-
T
tional homology spheres, where null–homologous means that the homology class
G
of a knot is trivial in the first homology group with Z-coefficients of the ambient
.
h space.
t Ifthe pair(Mˆ,Kˆ)ofa rationalhomologysphereMˆ andanull-homologousknot
a
m Kˆ in Mˆ is given by a finite cyclic branched cover over S3 branched along a knot
[ K, where Kˆ is the lift of K, then we can compute the Alexander polynomial of Kˆ
by using the well–known formula
3
3v ∆Kˆ(t)= Y ∆K(ξt) up to a factor ±ta(a∈Z)
8 ξ∈{x∈C|xk=1}
2 where k is the order of the covering transformation group, ξ runs all over the k-th
3
roots of unity and ∆ (t) is the Alexander polynomial of K. Such formulas have
. K
7 been investigated from the viewpoint of Reidemeister torsion for a long time. In
0 particular,V. Turaev gave a formula for the Alexander polynomial of Kˆ in a finite
1
cyclic branched cover over S3, and a generalization in the case of links in general
1
: three–dimensional manifolds (we refer to [Tur86, Theorems 1.9.2 and 1.9.3 ]).
v
The purpose of this paper is to provide the generalization of the above formula
i
X givingthe Alexander polynomialof a knotina finite cyclic branchedcoveroverS3
r to a formula for the twisted Alexander polynomial of finite abelian covers, which
a
is a special kind of Reidemeister torsion. Especially, we also consider the twisted
Alexanderpolynomialforalinkinathree–dimensionalmanifoldfromtheviewpoint
of Reidemeister torsion in the same way as V. Turaev. But to deal with finite
abelian covers beyond finite cyclic covers, we adopt the approach of J. Porti in
his work [Por04]. Porti gave a new proof of Mayberry–Murasugi’s formula, which
gives the order of the first homology group of finite abelian branched covers over
S3 branchedalong links, by using Reidemeister torsiontheory. We call the twisted
Alexander polynomial the polynomial torsion regarded as a kind of Reidemeister
torsion.
Date:January17,2012.
2000 Mathematics Subject Classification. Primary: 57M25,Secondary: 57M27.
Keywordsandphrases. Reidemeistertorsion;TwistedAlexanderpolynomial;branchedcover;
links;Homologyorientation.
1
In this paper, we are interested in the Reidemeister torsion for a finite sheeted
abeliancovering. Wearemainlyintestedinlinkexteriorsinhomologythree–spheres
andtheirabeliancovers. Ourmaintheorem(seeTheorem1)isstatedforanabelian
coverMˆ →M betweentwothree–dimensionalmanifoldswhoseboundaryisafinite
union of tori as follows
∆ϕMbˆ⊗ρb(t)=ǫ· Y∆M(ϕ⊗ρ)⊗ξ(t)
ξ Gˆ
∈
where ∆ϕMbˆ⊗ρb(t) and ∆M(ϕ⊗ρ)⊗ξ(t) are the signed twisted Alexander polynomials, Gˆ
isthesetofhomomorphismsfromthecoveringtransformationgroupGtothenon–
zero complex numbers and ǫ is a sign determined by the homology orientations of
Mˆ and M.
To be more precise, we need two homomorphisms of the fundamental group to
define the twisted Alexander polynomial of a manifold. The symbol ϕ denotes a
surjective homomorphism from π (M) to a multiplicative group Zn and ρ denotes
1
a representationofπ (M), i.e., a homomorphismfromπ (M)to a linearautomor-
1 1
phism group Aut(V) of some vector space V (see Section 3 for the definition of
the polynomial torsion). In the definition of the twisted Alexander polynomial of
Mˆ, we use the pull–backs ϕ and ρ of ϕ and ρ to π (Mˆ). The homomorphisms ϕ
1
and ξ determine variables inb the twbisted Alexander polynomial of Mˆ. In our main
theorem, we assume that the composition of ξ with the quotient homomorphism
π (M)→π (M)/π (Mˆ)≃G factors through homomorphism ϕ (see Section 4).
1 1 1
When we choose Mˆ →M as a finite cyclic cover of a knot exterior E of K in
K
S3, ϕistheabelianizationhomomorphismπ (E )→π (E )/[π (E ),π (E )]≃
1 K 1 K 1 K 1 K
Z and ρ is the one–dimensional trivial representation, Theorem 1 reduces to the
classicalformulaforthe AlexanderpolynomialofKˆ,whereKˆ istheliftoftheknot
in the finite cyclic branched cover over S3
∆Kˆ(t)= Y ∆K(ξt),
ξ x C xk=1
∈{ ∈ | }
up to a factor ±ta (a ∈ Z), where k is the order of π (M)/π (Mˆ). Our formula
1 1
also provides the Alexander polynomial of a link in finite abelian branched covers
over S3 branched along the link.
Organization
The outline of the paper is as follows. Section 2 deals with some reviews on
the sign–determined Reidemeister torsion for a manifold. In Section 3, we give
the definition of the polynomial torsion (the twisted Alexander polynomial) for a
manifold whose boundary is a finite union of tori. In Section 4, we consider the
polynomial torsion of finite abelian covering spaces (see Theorem 1).
Acknowledgments
The authors gratefully acknowledge the many helpful ideas and suggestions of
J. Porti during the preparation of the paper. They also wish to express their
thanks to C. Blanchet and S. Friedl for several helpful comments and for their
encouragement. Ourthanks alsogoto T.Morifuji,K.MurasugiandV. Maillotfor
giving us helpful informations and comments regarding this work. The paper was
conceived when J.D. visited the CRM. He thanks the CRM for hospitality. The
firstauthor(J.D.)ispartiallysupportedbytheFrenchANRprojectANR-08-JCJC-
0114-01. The second author (Y.Y.) is partially supported by the GCOE program
at Graduate School of Mathematical Sciences, University of Tokyo and Research
2
Fellowships of Japan Society for the promotion of Science for Young Scientists.
Y.Y. visited CRM and IMJ while writing the paper. He thanks CRM and IMJ for
theirhospitality. Theauthorsalsowouldliketothankthe refereeforhiscomments
and helpful remarks.
2. Preliminaries
2.1. The Reidemeister torsion. We review the basic notions and results about
thesign–determinedReidemeistertorsionintroducedbyV.Turaevwhichareneeded
inthispaper. DetailscanbefoundinMilnor’ssurvey[Mil66]andinTuraev’smono-
graph [Tur02].
Torsion of a chain complex. LetC =(0→C −d→n C −d−n−−→1 ···−d→1 C →0)be
n n 1 0
a chain complex of finite dimensio∗nal vector spaces ov−er a field F. Choose a basis
c(i) of C and a basis hi of the i-th homology group H (C ). The torsion of C
i i
∗ ∗
with respect to these choices of bases is defined as follows.
For each i, let bi be a set of vectors in C such that d (bi) is a basis of B =
i i i 1
im(d : C → C ) and let h˜i denote a lift of hi in Z = ker(d : C → C ).−The
i i i 1 i i i i 1
set of vectors d− (bi+1)h˜ibi is a basis of C . Let [d (bi+1)h˜ibi/ci]∈F− denote
i+1 i i+1 ∗
the determinant of the transition matrix between those bases (the entries of this
matrix are coordinates of vectors in d (bi+1)h˜ibi with respect to ci). The sign-
i+1
determined Reidemeister torsion of C (with respectto the bases c andh ) is the
∗ ∗
∗
following alternating product (see [Tur01, Definition 3.1]):
n
(1) Tor(C ,c∗,h∗)=(−1)|C∗|·Y[di+1(bi+1)h˜ibi/ci](−1)i+1 ∈F∗.
∗
i=0
Here
|C |=Xαk(C )βk(C ),
∗ ∗ ∗
k>0
where α (C )= i dimC and β (C )= i dimH (C ).
i Pk=0 k i Pk=0 k
The torsi∗onTor(C ,c ,h ) does not∗depend on the choice∗sof bi nor onthe lifts
∗ ∗
h˜i. Note that if C is∗acyclic (i.e. if H =0 for all i), then |C |=0.
i
∗ ∗
Torsion of a CW-complex. Let W be a finite CW-complex and (V,ρ) be a pair of
a vector space with an inner product over F and a homomorphism of π (W) into
1
Aut(V). The vector space V turns into a right Z[π (W)]-module denoted V by
1 ρ
using the right action of π (W) on V given by v·γ = ρ(γ) 1(v), for v ∈ V and
1 −
γ ∈π (W). The complex of the universal cover with integer coefficients C (W;Z)
1
∗ f
alsoinheritsaleftZ[π (W)]-module structureviatheactionofπ (W)onW asthe
1 1
f
covering group. We define the V -twisted chain complex of W to be
ρ
C∗(W;Vρ)=Vρ⊗Z[π1(W)]C∗(Wf;Z).
The complex C (W;V ) computes the V -twisted homology of W which is denoted
ρ ρ
∗
by H (W;V ).
ρ
Le∗t ei,...,ei be the set of i-dimensional cells of W. We lift them to the
(cid:8) 1 ni(cid:9)
universal cover and we choose an arbitrary order and an arbitrary orientation for
the cells e˜i,...,e˜i . If we choose an orthonormal basis {v ,...,v } of V, then
(cid:8) 1 ni(cid:9) 1 m
we consider the corresponding basis
ci = v ⊗e˜i,...,v ⊗e˜i,··· ,v ⊗e˜i ,...,v ⊗e˜i
(cid:8) 1 1 m 1 1 ni m ni(cid:9)
obafsCisi(oWf C;V(ρW) =;VV)ρ.⊗NZo[wπ1(cWho)]oCsi∗n(gWffo;rZe).acWh ieacablalstihsehibaosfisthce∗V=-t⊕wiicstiead hgeoommoelotrgiyc
ρ ρ
∗
3
H (W;V ), we can compute the torsion
i ρ
Tor(C (W;Vρ),c∗,h∗)∈F∗.
∗
We mainly consider the torsion of acyclic chain complexes C (W;V ), i.e., the
ρ
homology group H (W;V ) = 0. For acyclic chain complex C ∗(W;V ), this defi-
ρ ρ
∗ ∗
nition only depends on the combinatorial class of W, the conjugacy class of ρ, the
choices of c . The basis c for C (W;V ) depends on the following choices:
∗ ∗ ρ
∗
1. an order of cells {ei} and an orientation of each {ei};
j j
2. a lift e˜i of ei and;
j j
3. an orthonormalbasis of the vector space V.
We summarize the effect of changing these choices to Tor(C (W;V ),c ,∅) in the
ρ ∗
∗
following three remarks.
Remark 1. We have the same Tor(C (W;V ),c ,∅) for all orthonormal bases of
ρ ∗
∗
V since the effect of change of orthonormal bases in V is given by multiplying
the determinant of the bases change matrix with power of χ(W). If the Euler
characteristic χ(W) is zero, then we have the same torsion for any basis of V.
Remark 2. The torsion Tor(C (W;V ),c ,∅) depends on the choice of the lifts e˜i
ρ ∗ j
∗
under the action of π (W) by ρ. The effect of different lift of a cell is expressed as
1
the determinant of ρ(γ) for some γ in π (W). To avoid this problem, we often use
1
representations into SL(V).
Remark 3. To define the Reidemeister torsion, we order the cells {ei} and chose
j
an orientation of each ei, if we choose a different order and different orienta-
j
tions of cells, we could change the torsion sign. To remove this sign ambiguity,
that only occurs when m is odd, we use the fact that the sign of the torsions
Tor(C (W;R),c ,h ) and Tor(C (W;R),c ,h ) change in the same way.
∗R ∗R ∗ ∗
∗ ∗
Therefore we usually consider the torsion Tor(C (W;V ),c ,∅) up to the above
ρ ∗
∗
indeterminacy, namely up to a factor ±detρ(γ) for some γ in π (W).
1
WecanconstructtheadditionalsigntermreferredtoinRemark3asfollows. The
cells e˜i 06i6dimW,16j 6n are in one–to–one correspondence with the
(cid:8) j(cid:12) i(cid:9)
(cid:12)
cells of W, their order and orientation are induced an order and an orientation for
the cells ei 06i6dimW,16j 6n . Again, corresponding to these choices,
(cid:8) j(cid:12) i(cid:9)
we get a basis(cid:12) ci over R of C (W;R).
R i
Choose a homology orientation of W, which is an orientation of the real vector
space H (W;R) = H (W;R). Let o denote this chosen orientation. Pro-
Li>0 i
∗
vide each vector space H (W;R) with a reference basis hi such that the basis
i R
h0,...,hdimW of H (W;R) is positively oriented with respect to o. Compute
(cid:8) R R (cid:9)
the sign–determined R∗eidemeister torsion Tor(C (W;R),c ,h ) ∈ R of the re-
∗R ∗R ∗
∗
sulting based and homology based chain complex and consider its sign
τ =sgn(Tor(C (W;R),c ,h ))∈{±1}.
0 ∗R ∗R
∗
We define the sign–refined twisted Reidemeister torsion of W (with respect to
o) to be
(2) τ0m·Tor(C (W;Vρ),c∗,∅)∈F∗
∗
wherem=dimFV. ThissignrefinementalsoworksforthechaincomplexC (W;Vρ)
∗
with non–trivial homology group. When the dimension of V is even, we do not
needthesignrefinement,i.e., thetorsionTor(C (W;V ),c ,∅)isdeterminedupto
ρ ∗
∗
detρ(γ) for some γ in π (W).
1
One can prove that the sign–refinedReidemeister torsion is invariantunder cel-
lular subdivision, homeomorphism and simple homotopy equivalences. In fact, it
4
is precisely the sign (−1)C∗ in Equation (1) which ensures all these important
| |
invariance properties to hold (see [Tur02]).
3. Definition of the polynomial torsion
Inthis sectionwe define the polynomialtorsion. This givesa point ofview from
the Reidemeister torsion to polynomial invariants of topological space.
HereafterM denotesacompactandconnectedthree–dimensionalmanifoldsuch
that its boundary ∂M is empty or a disjoint union of b two–dimensionaltori:
∂M =T2∪...∪T2.
1 b
In the sequel, we denote by V a vector space over C and by ρ a representation
of π (M) into Aut(V), and such that detρ(γ)=1 for all γ ∈π (M).
1 1
Next we introduce a twisted chain complex with some variables. It will be done
by usinga Z[π (M)]–module with variablesto define anew twisted chaincomplex.
1
We regard Zn as the multiplicative group generated by n variables t ,...,t , i.e.,
1 n
Zn = t ,...,t |t t =t t (∀i,j)
(cid:10) 1 n i j j i (cid:11)
and consider a surjective homomorphism ϕ: π (M) → Zn. We often abbreviate
1
the n variables (t ,...,t ) to t and the rational functions C(t ,...,t ) to C(t).
1 n 1 n
When we consider the right action of π (M) on V(t)= C(t)⊗V by the tensor
1
representation
ϕ⊗ρ 1: π (M)→Aut(V(t)), γ 7→ϕ(γ)⊗ρ 1(γ),
− 1 −
we have the associated twisted chain C (M;V (t)) given by
ρ
∗
C∗(M;Vρ(t))=Vρ(t)⊗Z[π1(M)]C∗(Mf;Z)
where f ⊗v⊗γ ·σ is identified with fϕ(γ)⊗ρ(γ) 1(v)⊗σ for any γ ∈ π (M),
− 1
σ ∈ C (M;Z), v ∈ V and f ∈ C(t). We call this complex the V (t)-twisted chain
ρ
comple∗xfof M.
Definition 1. Fix a homology orientation on M. If C (M;V (t)) is acyclic, then
ρ
the sign–refined Reidemeister torsion of C (M;V (t)) ∗
ρ
∗
∆ϕM⊗ρ(t1,...,tn)=τ0m·Tor(C (M;Vρ(t)),c∗,∅)∈C(t1,...,tn)\{0}
∗
is called the polynomial torsion of M.
Observe that the sign–refined Reidemeister torsion ∆ϕM⊗ρ(t1,...,tn) is deter-
mined up to a factor tm1···tmn like the classical Alexander polynomial.
1 n
Example 1 (J.Milnor [Mil68], P.Kirk & C.Livingston[KL99]). Suppose that M
is the knot exterior E = S3\N(K) of a knot K in S3 where N(K) is an open
K
tubular neighbourhood of K.
Ifthe representationρ∈Hom(π (E );Q)is the trivialhomomorphismandϕ is
1 K
theabelianizationofπ (E ), i.e., ϕ: π (E )→H (E ;Z)≃hti,thenthe twisted
1 K 1 K 1 K
chain complex C∗(EK;Q(t)ρ) is acyclic and the Reidemeister torsion ∆ϕE⊗Kρ(t) is
expressed as a rational function which is the Alexander polynomial ∆ (t) divided
K
by (t−1) (see also [Tur01, Tur02]).
Example 2. Suppose now that M is the link exterior E = S3\N(L) of a link
L
L in S3. We suppose that L has n components, where n > 2. We denote by
µ the meridian of the i-th component. Consider the abelianization ϕ: π (E ) →
i 1 L
Zn defined by ϕ(µ ) = t . Let ρ: π (E ) → GL(1;C) = C \ {0} be the one–
i i 1 L
dimensional representation such that ρ(µ ) = ξ . Then the twisted chain complex
i i
5
C∗(EL;C(t)ρ) is acyclic and the Reidemeister torsion ∆ϕE⊗Lρ(t1,...,tn) is given by
(up to ±(ξ1−1t1)k1···(ξn−1tn)kn, ki ∈Z)
∆ϕE⊗Lρ(t1,...,tn)=∆L(ξ1−1t1,...,ξn−1tn)
where ∆ (t ,...,t ) is the Alexander polynomial of L.
L 1 n
4. Torsion for finite sheeted abelian coverings
4.1. Statement of the result. Let Mˆ be a finite sheeted abelian covering of M,
where M denotes a compact and connected three–dimensional manifold such that
its boundary ∂M is empty or a disjoint union of b two–dimensionaltori:
∂M =T2∪...∪T2.
1 b
We denote byp the induced homomorphismfromπ (Mˆ) to π (M) bythe covering
1 1
map Mˆ →M. The associated deck transformation group is a finite abelian group
G of order |G|. We endow the manifolds M and Mˆ with some arbitrary homology
orientations.
We have the following exact sequence:
(3) 1→π (Mˆ)−→p π (M)−→π G→1.
1 1
When we consider the polynomialtorsionfor Mˆ, we use the pull–back of homo-
morphisms of π (M) as homomorphisms of π (Mˆ). We denote by ϕ a surjective
1 1
homomorphism from π (M) to Zn and by ϕ the pull-back by p. We also suppose
1
that π factors through ϕ. Our situation is sbummarized as follows:
π1(Mˆ)◆◆ϕ◆bp◆∗◆◆◆// π◆''1(M(cid:15)(cid:15)(cid:15)(cid:15)ϕs) sπs s99//G
Zn
Similarly we use the symbol ρ for the pull–back of ρ: π (M) → Aut(V) by
1
p, where V is a vector space. Fbor homomorphisms of the quotient group G ≃
π (M)/π (Mˆ),weusethePontrjagindualofGwhichisthesetofallrepresentations
1 1
ξ: G → C = C\{0} from G to non–zero complex numbers. Let Gˆ denote this
∗
space.
We give the statement of the polynomial torsion for abelian coverings via that
of the based manifold.
Theorem 1. With the above notation, we suppose that the twisted chain com-
plex C (M;Vρ(t)) is acyclic. Then the twisted chain complex C (Mˆ;Vρb(t)) is also
acyclic∗and the polynomial torsion is expressed as ∗
(4) ∆ϕMbˆ⊗ρb(t)=ǫ· Y∆M(ϕ⊗ρ)⊗ξ(t)
ξ Gˆ
∈
where ǫ is a sign equal to τ (Mˆ)m·τ (M)mG and m=dimV.
0 0 | |
Remark 4. As we already observed, the sign term in Equation (4) is not relevant
when m is even.
Remark 5 (Explanation of Formula (4) with variables). If we denote by ξ¯ the
composition ξ◦π¯ as in the following commutative diagram
π1ϕ(M(cid:15)(cid:15)(cid:15)(cid:15) s)sssππs¯sss//99 G ξ //C
Zn
6
then Formula (4) can be written concretely as follows:
∆ϕMbˆ⊗ρb(t1,...,tn)=ǫ· Y∆ϕM⊗ρ(t1ξ¯(t1),...,tnξ¯(tn))
ξ Gˆ
∈
In the special case where n = 1, G = Z/qZ and Mˆ is the q–fold cyclic covering
M of M, then we have that ξ¯(t)=e2πk√ 1/q, for k =0,...,q−1. Hence we have
q −
the following covering formula for the polynomial torsion.
Corollary 1. Suppose that ϕ(π (M))= hti and ϕ(π (M ))= hsi⊂ hti, where we
1 1 q
suppose that s=tq. We have b
q 1
∆ϕMb⊗qρb(s)=∆ϕMb⊗qρ(tq)=ǫ· Y− ∆ϕM⊗ρ(e2πk√−1/qt).
k=0
The torsion ∆ϕMb⊗qρ(tq) in Corollary 1 can be regarded as a kind of the to-
tal twisted Alexander polynomial introduced in [SW09]. Hirasawa and Mura-
sugi [HM07] worked on the total twisted Alexander polynomial for abelian repre-
sentationsas inExample 1 andthey observedthe similar formulaas inCorollary1
intermsofthetotalAlexanderpolynomialandtheAlexanderpolynomialofaknot
in the cyclic branched coverings over S3.
4.2. Proof of Theorem 1. We use the same notation as in Remark 5.
First observe the following key facts:
• the universal cover M of M is also the one of Mˆ,
• the torsion ∆ϕMbˆ⊗ρ isfcomputed using the twisted complex
V (t ,...,t )⊗ C (M;Z),
ρ 1 n Z[π1(Mˆ)] ∗ f
• whereas the torsion ∆ϕM⊗ρ is computed using
Vρ(t1,...,tn)⊗Z[π1(M)]C∗(Mf;Z).
Lemma 2. Let x∈V (t ,...,t ), c∈C (M;Z). For γ ∈π (M), xγ 1⊗ γc
ρ 1 n ∗ f 1 − π1(Mˆ)
only depends on π(γ) ∈ G. For g ∈ G, choose γ ∈ π (M) such that π(γ) = g and
1
set
(5) g⋆(x⊗ c)=xγ 1⊗ γc.
π1(Mˆ) − π1(Mˆ)
This defines a natural action of G on V (t ,...,t )⊗ C (M;Z).
ρ 1 n π1(Mˆ) ∗ f
Further observe that, since for any lift γ of g, γ is not contained in p(π (Mˆ)),
1
we can not reduce the right hand side in Equation (5).
Proof. Take another lift γ in π (M) of g ∈ G. Since γ = γˆγ for some γˆ ∈
′ 1 ′
p(π (Mˆ)), we can see that xγ 1⊗ γ c=xγ 1⊗ γc. (cid:3)
1 ′− π1(Mˆ) ′ − π1(Mˆ)
The proof of Theorem 1 is based on the following technical lemma.
Lemma 3. The map
(6) Φ: Vρ(t)⊗Z[π1(Mˆ)]C∗(Mf;Z)→(Vρ(t)⊗CC[G])⊗Z[π1(M)]C∗(Mf;Z)
given by
Φ(x⊗ c)=(x⊗1)⊗c
π1(Mˆ)
is an isomorphism of complexes of C[G]-modules where the action of G on the
twisted complex (Vρ(t)⊗CC[G])⊗Z[π1(M)]C∗(Mf;Z) is given by
g·(x⊗g′⊗c)=x⊗gg′⊗c.
7
and the right action of γ ∈π1(M) on Vρ(t)⊗CC[G] is defined by
((f ⊗v)⊗g)·γ =fϕ(γ)⊗ρ−1(γ)(v)⊗π(γ)g,
where f ∈C(t), v ∈V and g ∈G.
Proof of Lemma 3. WefirstobservethatΦisawell–definedchainmapofC–vector
spaces since π (Mˆ) is a normal subgroup of π (M). By the definition, we can see
1 1
thatΦ(x⊗ γc)=Φ(xγ⊗ c)foranyγ inπ (Mˆ). HenceΦiswell–defined.
π1(Mˆ) π1(Mˆ) 1
FromΦ(∂(x⊗ c))=Φ(x⊗ ∂c)=(x⊗1)⊗∂c=∂((x⊗1)⊗c)=∂(Φ(x⊗c))
π1(Mˆ) π1(Mˆ)
it follows that Φ◦∂ =∂◦Φ. The G-equivariance of Φ follows from
Φ(g⋆(x⊗π1(Mˆ)c))=Φ(xγ−1⊗π1(Mˆ)γc)
=(xγ−1⊗1)⊗γc
=(x⊗g)⊗c
=g·Φ(x⊗ c).
π1(Mˆ)
We canprovethat Φ is anisomorphismby taking its inverseΨ as Ψ((x⊗g)⊗c)=
g⋆(x⊗c). (cid:3)
WementionbasesofthechaincomplexV (t)⊗ C (M;Z)beforethenext
ρ Z[π1(Mˆ)] ∗ f
step. The following basis
(7) ˆc∗ = [{xk⊗π1(Mˆ)γgeij|16j 6ni,g ∈G,16k 6m}
i>0 e
is the geometric basis used to compute the polynomial torsion ∆ϕMbˆ⊗ρb. When we
consider the bases change from the basis in Equation (7) to the basis in the next
equation
(8) {g⋆(x ⊗ ei)=x γ 1⊗ γ ei |16j 6n ,g ∈G,16k 6m},
k π1(Mˆ) j k g− π1(Mˆ) g j i
e e
we can see that the action of γg−1 arises the change in ∆ϕMbˆ⊗ρb by multiplying its
determinant powered the Euler characteristic of M. Since the Euler characteristic
of M is zero, the polynomial torsion ∆ϕMbˆ⊗ρb can also be computed using the basis
in Equation (8). Finally observe that Φ maps the basis in Equation (8) to the
geometric basis
(9) c∗G = [{(xk⊗g)⊗eij|16j 6ni,g ∈G,16k 6m},
i>0 e
thus
∆ϕMbˆ⊗ρb=τ0(Mˆ)m·Tor((Vρ(t)⊗CC[G])⊗Z[π1(M)]C∗(Mf;Z),ˆc∗,∅).
Now, we want to compute the torsion of (Vρ(t)⊗CC[G])⊗Z[π1(M)]C∗(Mf;Z) in
terms of polynomial torsions of M. To this end we use the decomposition along
orthogonal idempotents of the group ring C[G], see [Ser78] for details. Associated
to ξ ∈Gˆ, we define:
1
fξ = Xξ(g−1)g ∈C[G].
|G|
g G
∈
The properties of f are the following
ξ
fξ2 =fξ, fξfξ′ =0( if ξ 6=ξ′), Xfξ =1
ξ Gˆ
∈
and
g·f =ξ(g)f , for all g ∈G.
ξ ξ
8
We have the following C[G]-modules decomposition of the group ring as a direct
sum according to its representations:
(10) C[G]= MC[fξ].
ξ Gˆ
∈
Here each factor is the 1-dimensional C-vector space which is isomorphic to the
C[G]-module associated to ξ: G→C .
∗
Following[Por04,Section3],correspondingtothedecompositioninEquation(10)
we have a decomposition of complexes of C[G]-modules:
(Vρ(t)⊗CC[G])⊗Z[π1(M)]C∗(Mf;Z)= M(Vρ(t)⊗CC[fξ])⊗Z[π1(M)]C∗(Mf;Z).
ξ Gˆ
∈
iRseamcayrcklic6,.siTnhceisodneecocmanposseietiotnhaitmepalciehscthhaaitn(cVoρm(tp)le⊗xC(CVρ[G(t]))⊗⊗CZ[πC1[(fMξ]))] C⊗∗Z([πMf1(;MZ))]
C (M;Z) is acyclic from our assumptions and a change of variables.
∗ f
ThegeometricbasisinEquation(9)inducesabasiscompatiblewiththe decom-
position in Equation (10) by replacing {g|g ∈ G} by {f |ξ ∈ Gˆ}. The change
ξ
of bases cancels when we compute the torsion because Euler characteristic is zero,
see [Por04, Lemma 5.2]. And thus decomposition inEquation(10)implies that (in
the natural geometric bases):
∆ϕMbˆ⊗ρb=τ0(Mˆ)m·Tor((Vρ(t)⊗CC[G])⊗Z[π1(M)]C∗(Mf;Z),c∗G,∅)
(11) =τ0(Mˆ)m· YTor((Vρ(t)⊗CC[fξ])⊗Z[π1(M)]C∗(Mf;Z),c∗,∅).
ξ Gˆ
∈
Eachfactor in the right handside is relatedto the polynomialtorsionofM and
its relation is given by the following claim.
Lemma 4. We have:
∆M(ϕ⊗ρ)⊗ξ =τ0(M)m·Tor((Vρ(t)⊗CC[fξ])⊗Z[π1(M)]C∗(Mf;Z),c∗,∅).
Proof of Lemma 4. One canobserve that, as a Z[π1(M)]-module, Vρ(t)⊗CC[fξ] is
isomorphictoV (t)simplybyreplacingtheactionϕ⊗ρby(ϕ⊗ρ)⊗ξ. Thisproves
ρ
the equality of torsions. (cid:3)
Proof of Theorem 1. Combining Equation (11) and Lemma 4, we obtain
∆ϕMbˆ⊗ρb=τ0(Mˆ)m·τ0(M)m|G|· Y∆M(ϕ⊗ρ)⊗ξ
ξ Gˆ
∈
which achieves the proof of Formula (4). (cid:3)
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Institut de Math´ematiques de Jussieu, Universit´e Paris Diderot–Paris 7, UFR de
Math´ematiques,Case 7012,BaˆtimentChevaleret,75205Paris Cedex 13France
E-mail address: [email protected]
DepartmentofMathematics,TokyoInstituteofTechnology,2-12-1OokayamaMeguro-
ku,Tokyo 152-8551,Japan
E-mail address: [email protected]
10
