Monotonicity of Degrees of Generalized Alexander Polynomials of Groups and 3-Manifolds

0.31 MB·English

by  Shelly L. Harvey

MONOTONICITY OF DEGREES OF GENERALIZED
ALEXANDER POLYNOMIALS OF GROUPS AND 3-MANIFOLDS
5
0
0
SHELLYL.HARVEY†
2

n
a Abstract. We investigate the behavior of the higher-order degrees, δ¯n, of a
J finitelypresentedgroupG. Theseδ¯n arefunctionsfromH1(G;Z)toZwhose
2 values are the degrees certain higher-order Alexander polynomials. We show
1 that ifdef(G)≥1or G isthe fundamental groupofa compact, orientable 3-
manifoldthenδ¯nisamonotonicallyincreasingfunctionofnforn≥1. Thisis
] falseforgeneralgroups. Asaconsequence,weshowthatifa4-manifoldofthe
T formX×S1admitsasymplecticstructurethenX“looksalgebraicallylike”a
G 3-manifold that fibers over S1, supporting a positive answer to a question of
Taubes. ThisgeneralizesatheoremofS.Vidussi[V2]andisanimprovementon
.
h theresultsin[Ha1]. Wealsofindnewconditionsona3-manifoldXwhichwill
t guaranteethattheThurstonnormoff∗(ψ),forψ∈H1(X;Z)andf :Y →X
a
asurjectivemaponπ1,willbeatleastaslargetheThurstonnormofψ. When
m
X andY areknotcomplements,thisgivesapartialanswertoaquestionofJ.
[ Simon.
Moregenerally,wedefineΓ-degrees,δ¯Γ,correspondingtoasurjectivemap
1 G։Γforwhich Γis poly-torsion-free-abelian. Under certain conditions, we
v showtheysatisfyamonotonicityconditionifonevariesthegroup. Asaresult,
0 weshowthatthesegeneralizeddegreesgiveobstructionstothedeficiencyofa
9 group beingpositive and obstructions toa finitelypresented groupbeing the
1 fundamental groupofacompact, orientable3-manifold.
1
0
5
0 In [Ha1], we defined some new invariants δ¯n for a finite CW-complex X. These
/ invariants depended only on the fundamental group of X and measured the “size”
h
of the successive quotients of the rational derived series of π (X). Given X and a
t 1
a cohomologyclassψ ∈H1(X),δ¯ (ψ)wasdefinedtobethedegreeofa“higher-order
n
m
Alexander polynomial.” Although defined algebraically, these degrees have many
: topologicalapplicationsinthe casethatX is a3-manifold. Inthis case,we showed
v
i that the δ¯n give new estimates for the Thurston norm of a 3-manifold generaliz-
X
ing a theorem of C. McMullen [Mc]. Recall that the Thurston norm of a class
r ψ ∈ H1(X;Z), ||ψ|| , is defined to be the minimum negative euler characteristic
a T
ofall(possibly disconnected)surfacesF whose homologyclass[F]∈H (X,∂X;Z)
2
is Poincaredual to ψ and such that each component of F is non-positively curved.
The δ¯ also give new algebraic obstructions to a 3-manifold fibering over S1, to a
n
4-manifoldoftheformX×S1admittingasymplecticstructure,andtoa3-manifold
being Seifert fibered. They were also shown to have applications to minimal ro-
pelength and genera of knots and links in S3. Related work has been done by T.
Cochran, K. Orr, P. Teichner, for knots and knot concordance in [C] and [COT].
Recently, V. Turaev [T] has generalized some of the results in [Ha1].
Since δ¯ only depends on the fundamental group, we can consider δ¯ as an
n n
invariant of a general group G, δ¯ : H1(G;Z) → Z. In this paper, we continue to
n
†TheauthorwaspartiallysupportedbyanNSFPostdoctoral Fellowship.
1
2 SHELLYL.HARVEY
investigate the special behavior of the δ¯ when G is the fundamental group of a
n
3-manifold (with empty or toroidal boundary) or a group with deficiency at least
1. The results give new algebraic information about the topology of a symplectic
4-manifolds of the form X ×S1. They give obstructions to a finitely presented
group having positive deficiency or being the fundamental group of a compact,
orientable3-manifold(withorwithout boundary). They alsogivenew information
about the behavior of the Thurston norm under a map between 3-manifolds which
is surjective on π . We state some of our main theorems and their applications
1
below.
In [Ha1], we constructed examples of 3-manifolds for which δ¯ was a strictly
n
increasing function of n for n ≥ 0. Moreover, it was conjectured that the δ¯ were
n
always a monotonically increasing function of n for n ≥ 1. We show that this
conjecture is true. By δ¯ ≤ δ¯ (respectively δ¯ = 0) we mean that δ¯ (ψ) ≤
n n+1 n n
δ¯ (ψ) (respectively δ¯ (ψ)=0) for all ψ ∈H1(X).
n+1 n
Corollary2.10. LetX beaclosed, orientable, connected3-manifold. Ifβ (X)≥2
1
then
δ¯ ≤δ¯ ≤···≤δ¯ ≤··· .
0 1 n
If β (X) = 1 and ψ is a generator of H1(X) then δ¯ (ψ)− 2 ≤ δ¯ (ψ) ≤ ··· ≤
1 0 1
δ¯ (ψ)≤···.
n
AsaconsequenceofCorollary2.10,weshow(inTheorem3.8)thatif4-manifold
ofthe formX×S1 admits asymplectic structure thenX “looksalgebraicallylike”
a 3-manifold which fibers over S1, thus further supporting a conjecture of Taubes.
TheproofofTheorem3.8usesatheoremofVidussiin[V2]whoprovesthistheorem
in the case n=0.
Theorem 3.8. Let X be an closed, irreducible 3-manifold such that X×S1 admits
a symplectic structure. If β (X)≥2 there exists a ψ ∈H1(X;Z) such that
1
δ¯ (ψ)=δ¯ (ψ)=···δ¯ (ψ)=···=kψk .
0 1 n T
If β (X)=1 then for any generator ψ of H1(X;Z),
1
δ¯ (ψ)−2=δ¯ (ψ)=···δ¯ (ψ)=···=kψk .
0 1 n T
More generally, we define δ¯ (ψ) for any group G and any “admissible pair”
Γ
(φ : G ։ Γ,ψ : G ։ Z) of G. When G is a finitely presented group with
Γ
def(G)≥1, we show that the δ¯ satisfy a monotonicity condition. We also provea
n
similartheoremwhenGisthefundamentalgroupofaclosed,orientable3-manifold
(see Theorem 2.9).
Theorem 2.2. Let G be a finitely presented group with def(G)≥1 and (φ ,φ ,ψ)
Λ Γ
be an admissible triple for G. If (φ ,φ ,ψ) is not initial then
Λ Γ
(1) δ¯ (ψ)≥δ¯ (ψ)
Λ Γ
otherwise
(2) δ¯ (ψ)≥δ¯ (ψ)−1.
Λ Γ
Asaconsequenceofthemonotonicitytheorems,weseethattheδ¯ giveobstruc-
Γ
tions to the deficiency of a groupbeing positive or being the fundamental group of
acompact,orientable3-manifold. Theseobstructionsarenon-trivialevenwhenthe
groupsΓ andΛ are abelian. For example, we can easily recoverthe (known) result
MONOTONICITY OF THE DEGREES 3
that Zm cannot be the fundamental group of a compact 3-manifold when m ≥ 4
(see below or for more details see Example 3.2).
Proposition 3.1. Let G be a finitely presented group and (φ ,φ ,ψ) be an admis-
Λ Γ
sible triple for G.
(1) Suppose (φ ,φ ,ψ) is not initial. If δ¯ (ψ)<δ¯ (ψ) then def(G)≤0 and G
Λ Γ Λ Γ
cannot be the fundamental group of a compact, orientable 3-manifold (with
or without boundary).
(2) Suppose (φ ,φ ,ψ) is initial. If δ¯ (ψ)<δ¯ (ψ)−1 then def(G)≤0 and G
Λ Γ Λ Γ
cannot be the fundamental group of a compact, orientable 3-manifold with
at least one boundary component which is not a 2-sphere. In addition, if
δ¯ (ψ) < δ¯ (ψ)−2 then G cannot be the fundamental group of a compact,
Λ Γ
orientable 3-manifold (with or without boundary).
LetusconsiderthesimplestcasewhenΛistheabelianization(modulotorsion)of
GandΓ=Z. Inthiscase,δ¯Z(ψ)isequaltotherankofH1((XG)ψ;Z)asanabelian
group where (X ) is the infinite cyclic cover of X , a finite CW-complex with
G ψ G
π (X )=G,correspondingtoψ(aslongasthisnumberisfinite). Moreover,δ¯ (ψ)
1 G Λ
is equal the Alexander norm of ψ which depends only on ψ and the multivariable
Alexander polynomial of G. For example, then the Alexander polynomial of Zm is
1soδ¯Zm(ψ)=0foranyψ. Moreover,thefirsthomologyofanyinfinite cycliccover
of the m-torus is Zm−1 so δ¯Z(ψ) = m−1. Thus, as mentioned above, we see that
Zm cannot be the fundamental group of a compact 3-manifold.
Recall that the ith-order degree of a group δ¯(ψ) is a specific example of the
i
degree δ¯ (ψ). We give examples of finite 2-complexes X with β (X ) = 1 for
Γ n,g 1 n,g
n,g ≥ 1 such that the ith-order degrees for 0 ≤ i ≤ n−1 of X are “large” but
n,g
the nth-orderdegreeis 0. Thus the fundamentalgroupof these spaces cannothave
positivedeficiency norcanthey be the fundamentalgroupofa compact,orientable
3-manifold (see Proposition 2.5 and Example 3.4).
Theorem2.9 alsohas applications to the study of the behaviorof the genus of a
knot under a surjective map on π . The following question was asked by J. Simon
1
(see R. Kirby’s Problem List [Ki, Question 1.12(b)]).
Question1.12(b)of [Ki] (J.Simon). IfJ andK areknotsinS3 andf :S3\L→
S3\K is surjective on π , is g(L)≥g(K)?
1
The answer to the above question is known to be “yes” when δ (K) = 2g(K).
0
We strengthen this result to the case when δ (K)=2g(K)−1.
n
Corollary 3.12. Suppose J and K are knots in S3 such that there exists a sur-
jective homomorphism ρ : π (S3 \ L) ։ π (S3 \ K). If δ¯ (K) = 2g(K) or
1 1 0
δ¯ (K)=2g(K)−1 for some n≥1 then g(L)≥g(K).
n
We also prove this is the case if we replace the genus of a knot by the Thurston
norm. The following corollary is a generalization of the result due to Gabai [Ga]
that a degree one map f : X → Y between three manifolds gives the inequality
||f∗(ψ)|| ≥ ||ψ|| for all ψ ∈ H1(Y;Z). For simplicity, we state only the case
T T
when β (Y)≥2.
1
Corollary 3.11. Suppose there exists an epimorphism ρ : π (X)։ π (Y), where
1 1
X and Y are compact, orientable 3-manifolds, with toroidal or empty boundaries,
4 SHELLYL.HARVEY
such that β (X) = β (Y) ≥ 2 and r (X) = 0. Let ψ ∈ H1(π (Y);Z). If δ¯ (ψ) =
1 1 0 1 n
||ψ|| for some n≥0 then
T
||ρ∗(ψ)|| ≥||ψ|| .
T T
1. Definitions
We will define the higher-order degrees δ¯ and ranks r of a group G and sur-
Γ Γ
jective homomorphism φ : G ։ Γ. This definition will agree with the definition
Γ
of δ¯ given for a CW-complex X (as defined in §3 of [Ha1]) when G = π (X),
n 1
Γ = G/G(n+1) and φ = φ : G ։ G/G(n+1), the natural projection map. For
r Γ n r
more details see [Ha1, §3, §4 and §5] and [C, §2,§3,§5].
We recall the definition of a poly-torsion-free-abeliangroup.
Definition 1.1. A group Γ is poly-torsion-free-abelian(PTFA) if it admits a nor-
malseries{1}=G ⊳G ⊳···⊳G =Γ suchthat eachof the factors G /G is
0 1 n i+1 i
torsion-free abelian.
Remark 1.2. Recall that if A⊳G is torsion-free-abelianand G/A is PTFA then
GisPTFA.AnyPTFAgroupistorsion-freeandsolvable(theconverseisnottrue).
Also, any subgroup of a PTFA group is a PTFA group [P, Lemma 2.4, p.421].
Some examples of interesting series associated to a group G are the rational
lowercentralseriesofG(see Stallings[Sta]),the rationallowercentralseriesofthe
rational commutator subgroup of G, the rational derived series G(n) of G (defined
r
(n)
below), and the torsion-free derived series G of G (see [CH]). In this paper, our
H
examples and applications will use the rational derived series of a group (defined
below). We point out that the torsion-free derived series is very interesting since
it gives new concordance invariants of links in S3 (see [CH] or [Ha2]). For any
of the subgroups N in the above mentioned series, G/N is a PTFA group. In
particular,for eachn≥0, G/G(n+1) is PTFA by Corollary3.6 of [Ha1]. We recall
r
(n)
the definition of G .
r
Definition 1.3. Let G be a group and G(0) =G. For n≥1 define
r
G(n) = g ∈G(n−1) |gk ∈ G(n−1),G(n−1) for some k ∈Z−{0}
r r r r
n h i o
to be the nth term of the rational derived series of G.
R.StrebelshowedthatifGisthefundamentalgroupofa(classical)knotexterior
then the quotients of successive terms of the derived series are torsion-free abelian
[Str]. Hence for knot exteriors we have G(i) = G(i). This is also well known to be
r
true for free groups. Since any non-compact surface has free fundamental group,
this also holds for all orientable surface groups.
We make some remarks about PTFA groups. Recall that if Γ is PTFA then
ZΓ is an Ore domain and hence ZΓ embeds in it right ring of quotients K :=
Γ
ZΓ(ZΓ−{0})−1 which is a skew field. More generally, if S ⊆ R is a right divisor
set of a ring R then the right quotient ring RS−1 exists ([P, p.146] or [Ste, p.52]).
By RS−1 we mean a ring containing R with the property that
(1) Every element of S has an inverse in RS−1.
(2) Every element of RS−1 is of the form rs−1 with r∈R, s∈S.
MONOTONICITY OF THE DEGREES 5
IfRisanOredomainandSisarightdivisorsetthenRS−1isflatasaleftR-module
[Ste,PropositionII.3.5]. Inparticular,K isaflatleftZΓ-module. Moreover,every
Γ
finitely generated right module over a skew field is free and such modules have a
well defined rank, rank , which is additive on short exact sequences [Co1, p.48].
KΓ
Thus, if C is a non-negative finite chain complex of finitely generated free right
ZΓ-modules thenthe Eulercharacteristicχ(C)= ∞ (−1)irankC is defined and
i=0 i
is equalto ∞i=0(−1)irankKΓHi(C;KΓ). In this pPaper, we will repeatedly use this
fact about Pthe Euler characteristic.
Let ψ : G ։ Z be a surjective homomorphism. Note that we will always be
consideringZasthemultiplicativegroupZ=htigeneratedbyt. Wewishtodefine
δ¯ (ψ) as an non-negative integer. However, in order to do this, we need some
Γ
compatibility conditions on Γ and ψ.
Definition 1.4. LetGbeagroup,φ :G։Γ,andψ :G։ZwhereΓisaPTFA
Γ
group. We saythat(φ ,ψ) isanadmissible pairforG ifthere existsa surjection
Γ
α : Γ ։ Z such that ψ = α ◦φ . If α is an isomorphism then we say that
Γ,Z Γ,Z Γ Γ,Z
(φ ,ψ) is initial.
Γ
Let (φ ,ψ) be an admissible pair for G. We define Γ′ := ker(α ). It is clear
Γ Γ,Z
that (φ ,ψ) is initial if and only if Γ′ = 1. Since Γ is PTFA by Remark 1.2, Γ′ is
Γ
PTFA. Hence Γ′ embeds in its right ring of quotients which we call K . Moreover,
Γ
ZΓ′−{0}isknowntobearightdivisorsetofZΓ[P,p. 609]hencewecandefinethe
right quotient ring R := ZΓ(ZΓ′−{0})−1. After choosing a splitting ξ : Z → Γ,
Γ
we see that any element of RΓ can be written uniquely as tniki where t = ξ(1)
and ki ∈ KΓ. In this way, one sees that RΓ is isomorphic tPo the skew polynomial
ring K [t±1](see the proofof Proposition4.5 of [Ha1] for more details). Moreover,
Γ
theembedding g :ZΓ′ →K extendsto thisisomorphismR →K [t±1](herewe
ψ Γ Γ Γ
are identifying K and t0K ).
Γ Γ
The abelian group (G ) = kerφ /[kerφ ,kerφ ] is a right ZΓ-module via
Γ ab Γ Γ Γ
conjugation,
[g]γ =[γ−1gγ]
for γ ∈ Γ and g ∈ kerφ . Moreover, (G ) is a ZΓ′-module via the inclusion
Γ Γ ab
ZΓ′ ֒→ZΓ. Thus,(GΓ)ab⊗ZΓKΓ and(GΓ)ab⊗ZΓ′KΓ arerightKΓ andKΓ-modules
respectively.
Definition 1.5. Let G be a group and φ : G ։ Γ a coefficient system with Γ a
Γ
PTFA group . We define the Γ-rank of G to be
kerφ
rΓ(G)=rankKΓ(cid:18)[kerφ ,keΓrφ ] ⊗ZΓKΓ(cid:19).
Γ Γ
For a general group G and coefficient system φ , this rank may be infinite.
Γ
However, if G is finitely generated and φ is non-zero then by Proposition 2.11 of
Γ
[COT], r (G)≤β (G)−1 and hence is finite. In the case that φ is the zero map,
Γ 1 Γ
r (G)=β (G).
Γ 1
Definition1.6. LetGbeafinitelygeneratedgroupand(φ ,ψ)anadmissiblepair
Γ
for G. We define the Γ-degree of ψ to be
kerφ
δ¯Γ(ψ)=rankKΓ(cid:18)[kerφ ,keΓrφ ] ⊗ZΓ′ KΓ(cid:19)
Γ Γ
if r (G)=0 and δ¯ (ψ)=0 otherwise.
Γ Γ
6 SHELLYL.HARVEY
We remark that (GΓ)ab⊗ZΓ′ KΓ is merely (GΓ)ab⊗ZΓKΓ[t±1] viewed as a KΓ-
module. Since G is a finitely generated group, (GΓ)ab ⊗ZΓ KΓ[t±1] is a finitely
generatedK [t±1]-module. Moreover,since K [t±1] is a (noncommutative left and
Γ Γ
right) principal ideal domain, [Co2, 2.1.1, p.49], the latter is isomorphic to
⊕l K [t±1] hp (t)i⊕ K [t±1] rΓ(G)
i=1 Γ i Γ
(cid:14) (cid:0) (cid:1)
[J, Theorem 16, p.43]. Thus, (GΓ)ab⊗ZΓ′ KΓ is a finitely generated KΓ-module if
and only if r (G) = 0 . In particular, if r (G) = 0 then δ¯ (ψ) is the sum of the
Γ Γ Γ
degrees of the p (t). Therefore, δ¯ (ψ) as defined above is always finite.
i Γ
LetusconsiderthecasewhenΓ=Zm. LetX beaCW-complexwithπ (X)=G
1
andX betheregularZm-coverofX correspondingtoφ . Consideranadmissible
φΓ Γ
pair (φZm,ψ) for G. This is one such that ψ = ψ′ ◦ φΓ where ψ′ : Zm ։ Z.
In this case, H (X ;Z) = kerφ /[kerφ ,kerφ ] is a module over the Laurent
1 φΓ Γ Γ Γ
polynomialringwith m variables,Z[Zm]. Moreover,H (X ;Z)canbe considered
1 φΓ
asamodule overtheLaurentpolynomialringwithm−1variablesZΓ′ =Z[Zm−1].
Note that the m−1 variables in Z[Zm−1] correspond to a choice of basis elements
of Γ′ = ker(αZm,Z : Zm ։ Z). Therefore, as long as the rank of H1(XφΓ;Z) as
a Z[Zm]-module is 0, δ¯Zm(ψ) is equal to the rank of H1(XφΓ;Z) as a Z[Zm−1]-
module. In particular, when m = 1, δ¯Z(ψ) is equal the rank of H1(Xψ;Z) as an
abelian group whereX istheinfinitecovercorrespondingtoψ aslongasthisrank
ψ
isfinite(otherwiseδ¯Z(ψ)=0). WhenZmistheabelianizationofG,δ¯Zm(ψ)=δ¯0(ψ)
(see below forthe definition ofδ¯ ) is equalto the Alexander norm(see [Mc]for the
0
definition of the Alexander norm) of ψ by [Ha1, Proposition 5.12].
We now define the higher-order degrees and ranks associated to a group G. For
each n≥0, let Γ = G/G(n+1) where G(n+1) is the (n+1)st-term of the rational
n r r
derived series of G as defined in Definition 1.3. We define the nth-order rank of
X to be
r (X)=r (X).
n Γn
Next, we remark that if ψ ∈ H1(G;Z) ∼= Hom(G;Z), then ψ(Gr(n+1)) = 1. Hence
for each primitive ψ ∈H1(G;Z) the pair (φ ,ψ) is an admissible pair for G. For
Γn
primitive ψ, we define the nth-order degree of ψ to be
δ¯ (ψ)=δ¯ (ψ).
n Γn
Fornon-primitiveψ,thereisaprimitivecohomologyclassψ′ ∈H1(X;Z)suchthat
ψ =mψ′. Define δ¯ (ψ)=mδ¯ (ψ′).
n n
Thus,foreachgroupGandn≥0wehavedefinedafunctionδ¯ :H1(G;Z)→Z
n
which is “linear on rays through the origin”. We put a partial ordering on these
functions by δ¯ ≤δ¯ if δ¯(ψ)≤δ¯ (ψ) for allψ ∈H1(G;Z). Also, we say that δ¯ =0
i j i j i
provided δ¯(ψ)=0 for all ψ ∈H1(G;Z).
i
Suppose f : E ։ G is a surjective homomorphism and (φ ,ψ) is an admissible
Γ
pair for G. Then there is an induced admissible pair (φ ◦ f,ψ ◦ f) for E. In
Γ
particular,wecanspeakδ¯YE(ψ◦f). Whenwehavethissituation,unlessotherwise
Γ
noted, we will use this admissible pair induced by G. When there is no confusion,
we will suppress the f and just write (φ ,ψ) when we mean (φ ◦f,ψ◦f) or ψ
Γ Γ
when we mean ψ◦f.
In this paper, we will often use the notation r (X) and δ¯X(ψ) for X a CW-
complex and ψ an element of H1(X;Z) ∼= H1(Γπ (X);Z). ΓBy this, we mean
1
MONOTONICITY OF THE DEGREES 7
r (π (X))andδ¯π1(X)(ψ)foranadmissiblepair(φ ,ψ)forπ (X). Theseareequiva-
Γ 1 Γ Γ 1
lenttothehomologicaldefinitionsgivenin[Ha1]. Thatis,if(φ ,ψ)isanadmissible
Γ
pair for π (X) then H (X;K [t±1]) and H (X;K ) are right K and K -modules
1 1 Γ 1 Γ Γ Γ
respectively and since K and K [t±1] are flat left ZΓ-modules [Ste, Proposition
Γ Γ
II.3.5], we see that
r (X)=rank H (X;K )
Γ KΓ 1 Γ
and
δ¯Γ(ψ)=rankKΓH1(X;KΓ[t±1])
if r (X)=0 and δ¯ (ψ)=0 otherwise.
Γ Γ
2. Main Results
We seek to study the behavior of δ¯ (ψ) as n increases. More generally, we
n
would like to compare δ¯ as we vary the group Γ. We show that the δ¯ satisfy a
Γ Γ
monotonicity condition provided the groups satisfy a compatibility condition. We
describe this condition below.
Definition 2.1. LetGbe agroup,φ :G։Λ,φ :G։Γ,andψ :G։Zwhere
Λ Γ
Λ and Γ are PTFA groups. We say that (φ ,φ ,ψ) is an admissible triple for G
Λ Γ
if there exist surjections α : Λ ։ Γ and α : Γ ։ Z such that φ = α ◦φ ,
Λ,Γ Γ,Z Γ Λ,Γ Λ
ψ = α ◦φ , and α is not an isomorphism. If α is an isomorphism then we
Γ,Z Γ Λ,Γ Γ,Z
say that (φ ,φ ,ψ) is initial.
Λ Γ
Note that if (φ ,φ ,ψ) an admissible triple then (φ ,ψ) and (φ ,ψ) are both
Λ Γ Λ Γ
admissiblepairs. Hence,inthiscase,wecandefinebothδ¯ (ψ)andδ¯ (ψ). We note
Λ Γ
that (φ ,φ ,ψ) is initial if and only if (φ ,ψ) is initial. Moreover,(φ ,ψ) is never
Λ Γ Γ Λ
initial since Λ ։ Γ is not an isomorphism. We will show that δ¯ (ψ) ≥ δ¯ (ψ) as
Λ Γ
long as the triple is not initial. We point out that even if α is an isomorphism,
Λ,Γ
we can define both the Λ- and Γ-degrees and in this case δ (ψ)=δ (ψ)!
Γ Λ
We now proceed to state and prove the main theorems.
Theorem 2.2. Let G be a finitely presented group with def(G)≥1 and (φ ,φ ,ψ)
Λ Γ
be an admissible triple for G. If (φ ,φ ,ψ) is not initial then
Λ Γ
(3) δ¯ (ψ)≥δ¯ (ψ)
Λ Γ
otherwise
(4) δ¯ (ψ)≥δ¯ (ψ)−1.
Λ Γ
Before proving Theorem 2.2, we will state a Corollary of the theorem and make
some remarks about the deficiency hypothesis in the theorem. First, let Γ be
n
the quotient of G by the (n+1)st term of the rational derived series as in Defini-
tion1.3. Recallthatforanyψ ∈H1(G;Z),(φ ,ψ)isanadmissiblepair. Moreover,
Γn
(φ ,φ ,ψ)isanadmissibletripleunlessG(n+1) =G(n+2) whichisinitialifand
r r
Γn+1 Γn
only if β (G) = 1 and n = 0. Hence by Theorem 2.2 we see that the δ¯ are a
1 n
nondecreasing function of n (for n ≥ 1). This behavior was first established for
the fundamental groups of knot complements in S3 by T. Cochran in [C, Theorem
5.4]. Recall that δ¯ ≥ δ¯ (respectively δ¯ = 0) means that δ¯ (ψ) ≥ δ¯ (ψ)
n+1 n n n+1 n
(respectively δ¯ (ψ)=0) for all ψ ∈H1(G;Z).
n
8 SHELLYL.HARVEY
Corollary 2.3. Let G be a finitely presented group with def(G)≥1. If β (G)≥2
1
then
δ¯ ≤δ¯ ≤···≤δ¯ ≤··· .
0 1 n
If β (G) = 1 and ψ is a generator of H1(G;Z) then δ¯ (ψ)−1 ≤ δ¯ (ψ) ≤ ··· ≤
1 0 1
δ¯ (ψ)≤···.
n
Proof. LetψbeaprimitiveclassinH1(G;Z). WecanassumethatG(n+1) 6=G(n+2)
r r
sinceifG(n+1) =G(n+2)thenδ¯ (ψ)=δ¯ (ψ)(notethatinthecaseβ (G)=1and
r r n+1 n 1
n = 0, δ¯ (ψ) = δ¯ (ψ) ≥ δ¯ (ψ)−1 is also satisfied). Therefore T = (φ ,φ ,ψ)
1 0 0 Γn+1 Γn
is an admissible triple. As mentioned above, T is initial if and only if β (G) = 1
1
andn=0. Hence if β (G)=1 and n=0 then by Theorem2.2, δ¯ (ψ)≥δ¯ (ψ)−1.
1 1 0
Otherwise, δ¯ (ψ)≥δ¯ (ψ).
n+1 n
If β (G) ≥ 2 and ψ is not primitive then ψ = mψ′ for some primitive ψ′ and
1
m≥2. Hence, δ¯ (ψ)=mδ¯ (ψ′)≥mδ¯ (ψ′)=δ¯ (ψ). (cid:3)
n+1 n+1 n n
We now make some remarks about the condition def(G) ≥ 1. First, if G has
deficiency at least 2 then the results of Theorem 2.2 and Corollary2.3 hold simply
because all of the degrees are zero.
Remark 2.4. If G is a finitely presented group with def(G)≥2 and (φ ,ψ) is an
Γ
admissible pair for G then r (G)≥1 and hence δ¯ (ψ)=0.
Γ Γ
To see this, let X be a finite, connected 2-complex with one 0-cell x , m 1-
G 0
cells, r 2-cells where m−r ≥ 2 and G = π (X ,x ). Then H (X ,x ;K ) has
1 G 0 1 G 0 Γ
a presentation with m generators and r relations so rank H (X ,x ;K ) ≥ 2
KΓ 1 G 0 Γ
and hence r (G) = r (X ) = rank H (X ,x ;K )−1 ≥ 1 [Ha1, §4 and §5].
Γ Γ G KΓ 1 G 0 Γ
Therefore, δ¯ (ψ)=0 for all ψ ∈H1(G;Z).
Γ
However,if the deficiency of G is not positive, we can create an infinite number
of examples where the theorem is false! We construct finitely presented groups for
whichthedegreesare“large”upto(butnotincluding)thenth stagebutthedegree
atthe nth stageiszero! Forsimplicity, weonlydescribeexampleswhenβ (G)=1.
1
However, the reader should notice that the same type of behavior can be seen for
groups with β (G)≥2 using the same techniques.
1
Proposition 2.5. For each g ≥ 1 and n ≥ 1 there exist examples of finitely
presented groups G with def(G ) ≤ 0 and β (G ) = 1 such that δ¯ (ψ) = 2g,
n,g n,g 1 n,g 0
δ¯(ψ) = 2g −1 for 1 ≤ i ≤ n−1 and δ¯ (ψ) = 0 whenever ψ is a generator of
i n
H1(G ;Z).
n,g
Proof. We will construct these examples by adding relations to the fundamental
groupofafiberedknotcomplementGthatkillthegeneratorsthe G(n+1) G(n+2)⊗
Kn. Let G be the fundamental group of a fibered knot K in S3 of ge(cid:14)nus g ≥ 1
andn≥1. Since K isfibered, G(1) is free,soG(n+1)/G(n+2) =G(n+1)/G(n+2) and
r r
An = G(n+1) G(n+2)⊗ZΓ′ Kn is a finitely generatedfree right Kn-module of rank
n
2g−1. Let a(cid:14)1,...,a2g−1 be the generators of An. Since Kn is an Ore domain, we
can find k ∈K such that a k ∈G(n+1)/G(n+2)⊗1. Pick γ ,...,γ ∈G(n+1)
j n j j 1 2g−1
such that [γ ] = a k and let H = G/ < γ ,...,γ > and η : G ։ H. Note
j j j 1 2g−1
thatsinceanyknotgrouphasdeficiency1,H hasapresentationwithmgenerators
and m+2g−2 relations. Since γ ,...,γ ∈ G(n+1), we have an isomorphism
1 2g−1
G/G(n+1) ∼=H/H(n+1) ∼=H/Hr(n+1). Therefore,δ¯0H(ψ)=δ¯0G(ψ)=2gandδ¯iY(ψ)=
δ¯X(ψ)=2g−1 for 1≤i≤n−1.
i
MONOTONICITY OF THE DEGREES 9
Since G′ ։ H′, we have H′/H(n+1) ∼= G′/G(n+1) for 0 ≤ i ≤ n hence Kn =
KGn ∼= KHn. Moreover, since G(n+1) ։ H(n+1), the map G(n+1) G(n+2) ⊗Kn →
H(n+1) H(n+2)⊗K issurjective. ButthegeneratorsofA ares(cid:14)enttozerounder
n n
this ma(cid:14)p, so H(n+1)/H(n+2)⊗K =0. Finally, H(n+1) =H(n+1) so
n r
H(n+1) H(n+1) H(n+1)
Hr(n+2) ⊗Kn ∼= H(n+2) ⊗Kn ∼=(cid:18)H(n+2)(cid:30){Z-torsion}(cid:19)⊗Kn =0
r r
(see Lemma 3.5 of [Ha1] for the second isomorphism) hence δ¯ (ψ)=0. (cid:3)
n
We will now prove Theorem 2.2.
Proof of Theorem 2.2. Ifthe deficiencyofGisatleast2thenbyRemark2.4,allof
the degrees are zero hence the conclusions of the theorem are true. Now we prove
the case when def(G) = 1. We can assume that r (G) = 0, otherwise δ¯ (ψ) = 0
Γ Γ
andhence the statement of the theoremis true since δ¯ (ψ) is alwaysnon-negative.
Λ
Since G is finitely presented, there is a finite 2-complex X such that G = π (X)
1
and χ(X)=1−def(G)=0. Recall that X is obtained from the presentation of G
with deficiency 1 by starting with one 0-cell, attaching a 1-cell for each generator
and a 2-cell for each relation in the presentation of G. Since Γ ։ Z and φ is
Γ
surjective, H (X;K )=0 for i6=1,2 [COT, Proposition2.9]. Moreover,χ(X)=0
i Γ
implies that rank H (X;K ) = rank H (X;K ) = r (G) = 0 since the Euler
KΓ 2 Γ KΓ 1 Γ Γ
characteristic can be computed using K -coefficients as mentioned in §- 1. Since
Γ
r (X) = 0, it follows that r (X) = 0 [Ha2]. Replacing Γ by Λ in the above
Γ Λ
argument, it follows that rank H (X;K )=0.
KΛ 2 Λ
LetX betheinfinitecycliccoverofX correspondingtoψ. Thereisacoefficient
ψ
system for X , φ′ :π (X )։Γ′, given by restricting φ to π (X ). Moreover,as
KΓ-modules Hψ1(XΓ ;K1Γ) ∼=ψ H1(Xψ;KΓ) so H1(Xψ;KΓ) iΓs a fin1itelψy generated free
K -module of rank δ¯ (ψ) (similarly for Λ). Since Γ′ is PTFA (and hence ZΓ′ is an
Γ Γ
Oredomain),thereexistsawedgeofecirclesW andamapf :W →X suchthat
ψ
f :H (W;K )→H (X ;K )
∗ 1 Γ 1 ψ Γ
is an isomorphism. Here, the coefficient system on W is given by φ′ ◦f . By the
∗
Γ
proof of Lemma 2.1 in [COT], kerφ 6= kerψ if and only if φ′ ◦f is non-trivial.
Γ Γ ∗
Moreover, since W is a finite connected 2-complex with H (W) = 0, if kerφ 6=
2 Γ
kerψ then H (W;K )∼=Ke−1 [COT, Lemma 2.12]; otherwise H (W;K )∼=Ke.
1 Γ Γ 1 Γ Γ
Up to homotopy we can assume that W is a subcomplex of X by replacing
ψ
X with the mapping cylinder of f. Consider the long exact sequence of the pair
ψ
(X ,W) with coefficients in K :
ψ Γ
H (X ;K )→H (X ,W;K )→H (W;K )→H (X ;K ).
2 ψ Γ 2 ψ Γ 1 Γ 1 ψ Γ
Since X has no 3-cells, there is a cell complex, C (X;ZΓ), which has no 3-cells.
i
Therefore, TH (X;ZΓ), the ZΓ-torsion submodule of H (X;ZΓ), is zero. Now,
2 2
the kernel of the map H (X;ZΓ) → H (X;K ) is TH (X;ZΓ). Moreover, we
2 2 Γ 2
have shown that H2(X;KΓ) = 0 hence H2(X;ZΓ) = 0. Thus, H2(Xψ;KΓ) ∼=
H2(X;KΓ[t±1])∼=H2(X;ZΓ)⊗ZΓK[t±1]=0. Sincethelastarrowinthesequenceis
anisomorphism,H (X ,W;K )=0. OurgoalistoshowthatH (X ,W;K )=0.
2 ψ Γ 2 ψ Λ
Thenby analyzingthe long exactsequence of the pair (X ,W) with coefficients in
ψ
K , it will follow that H (W;K ) → H (X ;K ) is a monomorphism. We note
Λ 1 Λ 1 ψ Λ
that kerφΛ 6= kerφΓ implies that rankKΛH1(W;KΛ) = e−1 as above. Thus, if
10 SHELLYL.HARVEY
kerφ 6= kerψ then (assuming the monomorphism above) δ¯ (ψ) ≥ e−1 = δ¯ (ψ);
Γ Λ Γ
otherwise δ¯ (ψ)≥e−1=δ¯ (ψ)−1.
Λ Γ
Consider the relative chain complex of (X ,W) with coefficients in ZΓ′:
ψ
0→C (X ,W;ZΓ′)−∂−2Γ→′ C (X ,W;ZΓ′)→.
2 ψ 1 ψ
SinceW hasno2-cells,X hasno3-cells. ThereforeH (X ,W;ZΓ′)isZΓ′-torsion
ψ 2 ψ
free, so H (X ,W;K ) = 0 implies that H (X ,W;ZΓ′) = 0 and hence ∂Γ′ is
2 ψ Γ 2 ψ 2
injective.
Let A = ker(α : Λ′ ։ Γ′). Since A is a subgroup of a PTFA group, A
Λ,Γ|Λ′
is PTFA by Remark 1.2. If M is any right ZΛ′-module then M ⊗ZA Z has the
structure of a right ZΓ′-module given by
( σ⊗n)γ = σγ⊗n
X X
for any γ ∈Γ′. Moreover,one can check that C∗(Xψ,W;ZΛ′)⊗ZAZ is isomorphic
to C (X ,W;ZΓ′) as right ZΓ′-modules. Thus, after making this identification,
∗ ψ
∂Λ′ : C (X ,W;ZΛ′) → C (X ,W;ZΛ′) is injective by the following result of R.
2 2 ψ 1 ψ
Strebel.
Proposition 2.6 (R. Strebel, [Str] p. 305). Suppose Γ is a PTFA group and R
is a commutative ring. Any map between projective right RΓ-modules whose image
under the functor −⊗ is injective, is itself injective.
RΓ
Finally, since K is flat as a ZΛ′-module, H (X ,W;K )=0 as desired. (cid:3)
Λ 2 ψ Λ
SupposeΛandΓareabeliangroupsandGisthefundamentalgroupofacompact
orientable manifold with toroidal (or empty) boundary. In this case, it can easily
be shown,using the results in [Mc] and[Ha1], that the inequalities in Theorem2.2
(and Theorem 2.9 below) are in fact equalities for all ψ which lie in the cone of an
open face of the Alexander normball. We show below that evenin this case, there
are ψ for which the inequality in Theorem 2.2 is necessary.
Example 2.7. Let X be the exterior of the Borromean rings in S3 and let G
be the fundamental group of the X. A Wirtinger presentation of G is given by
x,y,z |[z,[x,y−1]],[y,[z,x−1]] (see [F, p.10] for a similar presentation). Thus,
(cid:10)there is an epimorphism f :G։(cid:11) hy,zi by sending x to 1. Let ψ(0,m,n) :G։Z be
the homomorphism defined by ψ(x)=1, ψ(y)=tm, ψ(z)=tn where gcd(m,n)=
1. Since f factors through ψ , the rank of H of the infinite cyclic cover of X
(0,m,n) 1
corresponding to ψ is non-zero (see, for example, [Ha, Proposition 2.2]). It
(0,m,n)
followsthatδ¯Z(ψ(0,m,n))=0. However,onecancomputetheAlexanderpolynomial
of X (from the presentation of G) to be ∆ = (x−1)(y−1)(z−1). Therefore,
X
δ¯Z3(ψ(0,m,n))=|m|+|n|>0.
Now we consider the case when G is the fundamental group of a closed 3-
manifold. In this case, the deficiency of G is 0 so Theorem 2.2 does not suffice to
provea monotonicity resultfor G. The proofthat the degreessatisfy a monotonic-
ity relation will use Theorem 2.2 for 2-complexes but will also use some additional
topology of the 3-manifold. Before stating the corresponding theorem for closed
3-manifolds, we introduce an important lemma which will be used in the proof of
Theorem 2.9.