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A VANISHING THEOREM FOR TWISTED ALEXANDER
POLYNOMIALS WITH APPLICATIONS TO SYMPLECTIC
4-MANIFOLDS
STEFAN FRIEDL AND STEFANO VIDUSSI
Abstract. We extend earlier results by the authors regarding twisted Alexander
polynomials of 3–manifolds. Together with work of Dani Wise our results imply
1that given any 3–manifoldN and any non–ﬁbered class inH (N;Z) there exists a
representation such that the corresponding twisted Alexander polynomial is zero.
This result allows us to completely classify symplectic 4–manifolds with a free circle
action, and to determine their symplectic cones.
1. Introduction and main results
A 3{manifold pair is a pair (N,ϕ) where N is a compact, orientable, connected
13–manifold with toroidal or empty boundary, and ϕ∈H (N;Z) = Hom(π (N),Z) is1
1a nontrivial class. We say that a 3–manifold pair (N,ϕ) bers over S if there exists
1 1a ﬁbrationp: N →S such that the induced mapp : π (N)→π (S ) =Z coincides 1 1
with ϕ.
Given a 3-manifold pair (N,ϕ) and an epimorphism α: π (N) → G onto a ﬁnite1
α 1group we can consider the twisted Alexander polynomial ∆ ∈Z[t ], whose deﬁni-
N,ϕ
tion is summarized in Section 2. It is well–known that the twisted Alexander polyno-
1mials of a ﬁbered class ϕ∈H (N) are monic and that their degrees are determined
by the Thurston norm. In [FV11a] the authors showed that this condition is in fact
1suﬃcient to determine ﬁberedness. More precisely, if a nontrivial class ϕ∈H (N) is
αnot ﬁbered, then there exists a twisted Alexander polynomial ∆ that fails to beN,ϕ
monic or to have correct degree. We refer to Theorem 3.1 for the exact statement.
Inpreviouswork(see[FV08b])theauthorsdiscussedhowastrongerresult, namely
αthe vanishing of some twisted Alexander polynomial ∆ , would follow assumingN,ϕ
appropriate separability conditions for the fundamental group of N. In this paper,
using work of Wilton and Zalesskii, we will improve the result of [FV08b] by reducing
the separability condition to the hyperbolic pieces of N.
Before we state our main theorem we start with some deﬁnitions. First, recall that
a group π is called locally extended residually nite (or LERF for short) if for any
ﬁnitely generated subgroup A ⊂ π and any g ∈ π\A there exists an epimorphism
Date: December 21, 2011.
S. Vidussi was partially supported by NSF grant #0906281.
12 STEFAN FRIEDL AND STEFANO VIDUSSI
α: π → G to a ﬁnite group G such that α(g) ̸∈ α(A). Groups which are LERF are
also often referred to as being subgroup separable. Second, we introduce the following
De nition. A compact, connected, orientable 3-manifold N with empty of toroidal
boundary is called perfect if one of the following holds:
(1) N is reducible,
(2) N is irreducible, and for all hyperbolic piecesN in the JSJ decomposition wev
have either
(a) π (N ) is LERF, or1 v
(b) b (N ) = 1 and N is ﬁbered.1 v v
Note that irreducible graph manifolds by deﬁnition do not contain any hyperbolic
components in the JSJ decomposition, hence are perfect.
We can now state our main theorem:
1Theorem 1.1. Let (N,ϕ) be a 3{manifold pair with N perfect. If ϕ ∈ H (N) is
non bered, there exists an epimorphism α: π (N) → G onto a nite group G such1
that
α∆ = 0.N,ϕ
It has been a long standing conjecture that fundamental groups of hyperbolic 3–
manifolds are LERF. Recently Dani Wise has made remarkable progress towards an
aﬃrmative answer. The following theorem combines the statements of Theorems
14.1, 16.1 together with Corollary 14.16 of Wise [Wi11a]. (We refer to [Wi09, Wi11a,
Wi11b] for background material, deﬁnitions and further information.
Theorem 1.2. (Wise) If N is either a closed hyperbolic 3-manifold which admits a
geometrically nite surface or if N is a hyperbolic 3-manifold with nontrivial bound-
ary, then π (N) is virtually compact special.1
The precise deﬁnition of ‘virtually compact special’ is of no concern to us. What is
important is that it is well known that Wise’s theorem combined with previous work
of various authors implies the following corollary:
Corollary 1.3. Let N be a 3{manifold with b (N)≥ 1; then N is perfect.1
We refer to Section 5 for details and for precise references. In light of Corollary
1.3, Theorem 1.1 greatly strengthens the ‘if’ direction of Theorem 3.1.
In Section 6 we will see that the combination of Theorem 1.1 with work of Goda
and Pajitnov [GP05] implies a result on Morse-Novikov numbers of multiples of a
given knot. Furthermore, we will see that the combination of Theorem 1.1 with
work of Silver and Williams [SW09b] gives rise to a ﬁbering criterion in terms of the
number of ﬁnite covers of the ϕ-cover of N. Arguably, however, the most interesting
application of Theorem 1.1 is contained in Section 7, and regards the study of closed
4-manifolds with a free circle action which admit a symplectic structure. The main
result of Section 7 is then the proof of the ‘(1) implies (3)’ part of following:A VANISHING THEOREM FOR TWISTED ALEXANDER POLYNOMIALS 3
1Theorem 1.4. Let N be a closed perfect 3-manifold and let p: M → N be an S {
2 1bundle over N. We denote by p : H (M;R)→H (N;R) the map which is given by
2integration along the ber. Let ψ∈H (M;R). Then the following are equivalent:
(1) ψ can be represented by a symplectic structure,
1(2) ψ can be represented by a symplectic structure which is S {invariant,
2 1(3) ψ > 0 and ϕ = p (ψ) ∈ H (N;R) lies in the open cone on a bered face of
the Thurston norm ball of N.
(See Section 7 for details on the other implications.)
The implication ‘(1) implies (3)’ had already been shown to hold in the following
cases:
(1) for reducible 3–manifolds by McCarthy [McC01],
(2) if N has vanishing Thurston norm by Bowden [Bow09] and [FV08c], or if N
is a graph manifold, [FV08c],
(3) if the canonical class of the symplectic structure is trivial, [FV11c],
1 1(4) if M is the trivial S -bundle over N, i.e. the case that M = S ×N, see
[FV11a] for details.
Remark. (1) This paper can be viewed as the (presumably) last paper in a long
sequence of papers [FV08a, FV08b, FV08c, FV11a, FV11b, FV11c] by the
authorsontwistedAlexanderpolynomials,ﬁbered3-manifoldsandsymplectic
structures.
(2) Some steps in the proof of Theorem 1.4 (notably Propositions 7.2 and 7.3)
already appeared in an unpublished manuscript by the authors (see [FV08c]).
Acknowledgment. We wish to thank Henry Wilton for very helpful conversations.
Convention. Unless it says speciﬁcally otherwise, all groups are assumed to be
ﬁnitelygenerated,allmanifoldsareassumedtobeorientable,connectedandcompact,
and all 3-manifolds are assumed to have empty or toroidal boundary.
2. Definition of twisted Alexander polynomials
In this section we quickly recall the deﬁnition of twisted Alexander polynomial.
ThisinvariantwasinitiallyintroducedbyLin[Li01],Wada[Wa94]andKirk–Livingston
[KL99]. We refer to [FV10a] for a detailed presentation.
1Let X be a ﬁnite CW complex, let ϕ ∈ H (X;Z) = Hom(π (X),Z) and let1
α: π (X) → GL(n,R) be a representation over a Noetherian unique factorization1
domain R. In our applications we will take R =Z or R =Q. We can now deﬁne a
n 1 n 1leftZ[π (X)]–module structure on R ⊗ Z[t ] =:R [t ] as follows:1 Z
ϕ(g)g·(v⊗p) := (α(g)·v)⊗(t p),
n 1 n 1where g∈π (X),v⊗p∈R ⊗ Z[t ] =R [t ]. Put diﬀerently, we get a represen-1 Z
1tation α⊗ϕ: π (X)→ GL(n,R[t ]).1̸
4 STEFAN FRIEDL AND STEFANO VIDUSSI
eDenotebyX theuniversalcoverofX. Lettingπ =π (X),weusetherepresentation1
n 1 eα⊗ϕ to regard R [t ] as a left Z[π]–module. The chain complex C (X) is also a
1leftZ[π]–module via deck transformations. Using the natural involution g →g on
ethe group ring Z[π] we can view C (X) as a right Z[π]–module. We can therefore
consider the tensor products
ϕ
α n 1 n 1˜C (X;R [t ]) :=C (X)⊗ R [t ], Z[π (X)] 1
1 1which form a complex of R[t ]-modules. We then consider the R[t ]–modules
ϕ
α n 1 ϕ
α n 1H (X;R [t ]) :=H (C (X;R [t ])).
If α and ϕ are understood we will drop them from the notation. Since X is com-
1 1pact and since R[t ] is Noetherian these modules are ﬁnitely presented over R[t ].
We now deﬁne the twisted Alexander polynomial of (X,ϕ,α) to be the order of
n 1H (X;R [t ]) (see [FV10a] and [Tu01, Section 4] for details). We will denote it1
α 1 α 1as ∆ ∈ R[t ]. Note that ∆ ∈Z[t ] is well-deﬁned up to multiplication by aX,ϕ X,ϕ
1unit inR[t ]. We adopt the convention that we dropα from the notation ifα is the
trivial representation to GL(1,Z).
If α: π (N) → G is a homomorphism to a ﬁnite group G, then we get the reg-1
ular representation π (N) → G → Aut (Z[G]), where the second map is given by1 Z
left multiplication. We can identify Aut (Z[G]) with GL(|G|,Z) and we obtainZ
αthe corresponding twisted Alexander polynomial ∆ . We will sometimes writeN,ϕ
1 jGj 1H (X;Z[G][t ]) instead of H (X;Z [t ]).
The following lemma is well-known (see e.g. [Tu01, Remark 4.5]).
Lemma 2.1. Let (N,ϕ) be a 3{manifold pair and let α: π (N)→G be a homomor-1
α 1 1phism to a nite group. Then ∆ = 0 if and only if H (N;Z[G][t ]) is Z[t ]-1N,ϕ
torsion.
We will later need the following well-known lemma.
Lemma2.2. Let (N,ϕ) be a 3{manifold pair. Letα: π (N)→G andβ: π (N)→H1 1
be homomorphisms to nite groups such that Ker (α) ⊂ Ker(β). Then there exists
1p∈Q[t ] such that
βα 1∆ = ∆ ·p∈Q[t ].N,ϕ N,ϕ
β αIn particular, if ∆ = 0, then ∆ = 0.N,ϕ N,ϕ
Proof. We denote by α also the regular representation π (N) → Aut (Q[G]), and1 Q
similarly we byβ the regular representationπ (N)→ Aut (Q[H]). Note that1 Q
theassumptionKer(α)⊂ Ker(β)impliesthereexistsanepimorphismγ: G→H such
that β =γ◦α. Note that γ endowsQ[H] with the structure of a leftQ[G]-module.
It follows from Maschke’s theorem (see [La02, Theorem XVIII.1.2]) that there exists
a leftQ[G]-module P and an isomorphism of leftQ[G]-modules
∼Q[G] Q[H]⊕P.≠
̸
̸
A VANISHING THEOREM FOR TWISTED ALEXANDER POLYNOMIALS 5
α ∼We now denote by ρ the representation π (N)−→G→ Aut(P) GL(dim(P),Q). It=1
now follows from the deﬁnitions that
β ρα∆ = ∆ ·∆ .N,ϕ N,ϕ N,ϕ
3. Twisted Alexander polynomials and fibered 3{manifolds
Let (N,ϕ) be a 3–manifold pair. We denote by∥ϕ∥ the Thurston norm of a classT
1 1ϕ ∈ H (N;Z): we refer to [Th86] for details. We say that p(t) ∈ Z[t ] is monic
1if its top coeﬃcient equals ±1 and given a nonzero polynomial p(t) ∈ Z[t ] with
∑l ip = at,a = 0,a = 0 we deﬁne deg(p) =l−k.i k li=k
ItisknownthattwistedAlexanderpolynomialsgivecompleteﬁberingobstructions.
In fact the following theorem holds:
1 2 1 2Theorem 3.1. Let (N,ϕ) be a 3{manifold pair where N =S ×S ,S ×D . Then
(N,ϕ) is bered if and only if for any epimorphism α: π (N)→G onto a nite group1
α 1the twisted Alexander polynomial ∆ ∈Z[t ] is monic andN,ϕ
αdeg(∆ ) =|G| ·∥ϕ∥ +(1+b (N))divϕ ,T 3 αN,ϕ
where ϕ denotes the restriction of ϕ: π (N) → Z to Ker(α), and where we denoteα 1
by divϕ ∈N the divisibility of ϕ , i.e.α α
divϕ = max{n∈N|ϕ =nψ for some ψ: Ker(α)→Z}.α α
The‘onlyif’directionhasbeenshownatvariouslevelsofgeneralitybyCha[Ch03],
Kitano and Morifuji [KM05], Goda, Kitano and Morifuji [GKM05], Pajitnov [Pa07],
Kitayama [Kiy08], [FK06] and [FV10a, Theorem 6.2]. The ‘if’ direction is the main
result of [FV11a]. We also refer to [FV11b] for a more leisurely approach to the proof
of the ‘if’ direction.
4. The proof of Theorem 1.1
In this section we will prove Theorem 1.1. The approach we follow is the one we
used in [FV08b] to cover the case of a 3–manifold with certain subgroup separability
properties, adding as new ingredient the work of Wilton and Zalesskii [WZ10] (which
builds in turn on work of Hamilton [Ham01]), to cover the case where the 3–manifold
has a nontrivial JSJ decomposition. We start with the following.
Lemma 4.1. Let N be an irreducible 3-manifold with JSJ pieces N ,v ∈ V. Letv
α : π (N ) → G ,v ∈ V be homomorphisms to nite groups. Then there exists anv 1 v v
epimorphism β: π (N)→G to a nite group, such that Ker (β)∩π (N )⊂ Ker(α )1 1 v v
for all v∈V.6 STEFAN FRIEDL AND STEFANO VIDUSSI
Proof. We write K := Ker(α ),v ∈ V. Evidently there exists an n ∈ N with thev v
following property: for each v∈V and each boundary torus T of N we havev
n·π (T)⊂π (T)∩K .1 1 v
2∼(Here n·π (T) denotes the unique characteristic subgroup of π (T) Z of index=1 1
2n .) By [WZ10, Theorem 3.2] (which relies strongly on Lemmas 5 and 6 of [Ham01])
there exists an m ∈N and ﬁnite index normal subgroups L ⊂ π (N ),v ∈ V suchv 1 v
that for each v∈V and each boundary torus T of N we havev
nm·π (T) =π (T)∩L .1 1 v
WenowdeﬁneM =K ∩L ,v∈V. NotethatforeachJSJtorusT andtwoadjacentv v v
JSJ pieces the corresponding M-groups intersect to nm·π (T). It now follows from1
a standard argument (see e.g. [WZ10, Proof of Theorem 3.7]) that there exists a
ﬁnite index normal subgroupM ofπ (N) such thatM∩π (N )⊂M for anyv∈V.1 1 v v
Clearly the epimorphism π (N)→π (N)/M has the desired properties. 1 1
For the reader’s convenience we recall the statement of Theorem 1.1.
1Theorem 4.2. Let (N,ϕ) be a 3{manifold pair with N perfect. Then if ϕ∈H (N)
is non bered, there exists an epimorphism α: π (N)→G onto a nite group G such1
that
α∆ = 0.N,ϕ
Remark. As mentioned above, a proof of this theorem appears in [FV08b] for a 3–
manifold N in the following two cases:
(1) if the subgroups carried by Thurston norm minimizing surfaces are separable,
or
(2) if N is a graph manifold.
Recent work of Przytycki and Wise [PW11, Theorem 1.1] shows that the separability
condition (1) is in fact satisﬁed for all graph manifolds, which gives an alternative
proof for (2).
Proof. If N is reducible, the statement is proven to hold in [FV11a, Lemma 7.1].
Hence we reduce ourselves to the case whereN is irreducible. We denote by{N }v v2V
the set of JSJ components ofN and by{T } the set of JSJ tori in the JSJ decom-e e2E
1position of N. Let ϕ∈H (N;Z) be a nonﬁbered class. If the restriction of ϕ to one
of the JSJ tori is trivial, then it follows from [FV08b, Theorem 5.2] that there exists
αan epimorphism α: π (N) → G onto a ﬁnite group G such that ∆ = 0. We will1 N,ϕ
henceforth assume that the restriction of ϕ to all JSJ tori is nontrivial.
Given v ∈ V we denote by ϕ the restriction of ϕ to N . Since ϕ is nonﬁbered itv v
follows from [EN85, Theorem 4.2] that there exists a w ∈ V, such that (N ,ϕ ) isw w
not ﬁbered. If N is hyperbolic, then π (N ) is LERF since we assumed that N isw 1 w
perfect. The same holds if N is Seifert ﬁbered, by Scott’s theorem (see [Sc78]). Byw
[FV08b, Theorem 4.2] there exists in either case an epimorphism α : π (N )→Gw 1 w wA VANISHING THEOREM FOR TWISTED ALEXANDER POLYNOMIALS 7
αwonto a ﬁnite group G such that ∆ = 0. By Lemma 4.1 above there existsw N ,ϕw w
a homomorphism β: π (N) → G to a ﬁnite group, such that Ker(β)∩π (N ) =1 1 w
Ker(β )⊂ Ker(α ). (Here, given v∈V we denote by β the restriction of β to N .)w w v v
βwBy Lemma 2.2 we also have ∆ = 0.N ,ϕw w
Now, there exists a Mayer–Vietoris type long exact sequence of twisted homology
groups:
⊕ ⊕
1 1 1···→ H (T ;Z[G][t ])→ H (N ;Z[G][t ])→H (N;Z[G][t ])→....i e i v i
e2E v2V
RecallthatweassumedthattherestrictionofϕtoT isnontrivialforeache. Itfollowse
1from a straightforward calculation (see e.g. [KL99, p. 644]) that H (T ;Z[G][t ]) isi e
β1 vZ[t ]-torsion for each i and each e ∈ E. As ∆ = 0, Lemma 2.1 implies thatN ,ϕw w
1 1H (N ;Z[G][t ]) is not Z[t ]-torsion. But then it follows from the above exact1 w
β1 1sequence that H (N;Z[G][t ]) is notZ[t ]-torsion either, i.e. ∆ = 0. 1 N,ϕ
5. Wise’s results
As we already mentioned in the introduction, Wise recently set forth a proof of the
following theorem:
Theorem 5.1. (Wise) If N is either a closed hyperbolic 3-manifold which admits a
geometrically nite surface or if N is a hyperbolic 3-manifold with nontrivial bound-
ary, then π (N) is virtually compact special.1
The following is now a well-known consequence of Theorem 5.1.
Proposition 5.2. Let N be a hyperbolic 3{manifold with b (N) ≥ 1, then N is1
perfect.
We refer to [Wi11a, Corollary 14.3] for details in the closed case and we refer to
[AFW12] for a summary in the cusped case. For the reader’s convenience we quickly
recall the main steps of the proof in the closed case. Let N be a closed hyperbolic
3–manifold with b (N) ≥ 1. If b (N) = 1 and N is ﬁbered, then N is perfect by1 1
deﬁnition. Otherwise a straightforward Thurston norm argument (see [Th86]) shows
that N admits an incompressible surface Σ which does not lift to a ﬁber in any
ﬁnite cover. By a result of Bonahon and Thurston (see [Bon86]) the surface Σ is a
geometrically ﬁnite surface. By Theorem 5.1 the group π (N) is virtually compact1
special. It now follows from the work of Haglund and Wise [HW08] and Haglund
[Hag08, Theorem F] and the solution of the tameness conjecture due to Agol [Ag07]
and Calegari-Gabai [CG06] that π (N) is LERF.1
The following is now an immediate corollary to Proposition 5.2.
Corollary 5.3. Let N be a 3-manifold with b (N)≥ 1; then N is perfect.18 STEFAN FRIEDL AND STEFANO VIDUSSI
Proof. If N is reducible, it is perfect by deﬁnition. If N is an irreducible 3-manifold
withb (N)≥ 1,everyhyperbolicJSJcomponentofN haspositiveﬁrstBettinumber.1
The statement now follows immediately from Proposition 5.2.
6. Applications to 3{manifold topology
3 3Let K ⊂ S be a knot. We denote by X := S \νK the exterior of K. AK
regular Morse function on X is a function f: X → K such that all singularitiesK K
are nondegenerate and which restricts on the boundary ofX to a ﬁbration. Given aK
Morse map f we denote by m (f) the number of critical points of index i. A regulari
1 function f: X → S is called minimal if m (f) ≤ m (g) for anyK i i
Morse map g. It is shown by Pajitnov, Rudolph and Weber [PRW02] that any knot
admits a minimal regular Morse function. Its number of critical points is called the
Morse-Novikov number of K and denoted byMN(K). Note thatK is ﬁbered if and
only if MN(K) = 0. It is known that MN(K #K )≤MN(K )+MN(K ) (see1 2 1 2
[PRW02, Proposition 6.2]), but it is not known whether equality holds. It is not even
known whetherMN(n·K) =n·MN(K). The following theorem can be viewed as
evidence to an aﬃrmative answer for the latter question.
3Theorem6.1. LetK ⊂S be a non bered knot, such that X is perfect. Then thereK
exists a λ> 0, such that
MN(n·K)≥n·λ.
ThistheoremisanimmediateconsequenceofTheorem1.1andresultsofGodaand
Pajitnov [GP05, Theorem 4.2 and Corollary 4.6]. The statement is similar in spirit to
a result by Pajitnov (see [Pa10, Proposition 4.2]) on the tunnel number of multiples
ofa givenknot. As weexplained in Corollary 1.3, Wise’s results implythat the above
theorem applies to all knots.
Now let (N,ϕ) be a ﬁbered 3–manifold pair. In that case Ker(ϕ: π (N) → Z) is1
the fundamental group of a surface, in particular it is a ﬁnitely generated group, it
followsthattheϕ-coverofN admitsonlycountablymanyﬁnitecovers. Thefollowing
theorem,whichfollowsfromcombiningTheorem1.1withworkofSilverandWilliams
(see[SW09a,SW09b]),saysthattheconversetotheabovestatementholdsforperfect
3–manifolds.
1Theorem 6.2. Let N be a perfect 3-manifold and let ϕ∈H (N) = Hom(π (N),Z).1
1If (N,ϕ) does not ber over S , then theϕ-cover ofN admits uncountably many nite
covers.
For knot exteriors this result is an immediate consequence of Theorem 1.1 and a
result of Silver and Williams [SW09b, Theorem 3.4] for knots (see also [SW09a]). It
isstraightforwardtoverifythattheargumentbySilverandWilliamsalsocarriesover
to the general case./
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A VANISHING THEOREM FOR TWISTED ALEXANDER POLYNOMIALS 9
Note that Theorem 6.2 can be viewed as a signiﬁcant strengthening of Stallings’
1ﬁbering theorem (see [St62]), which says that a classϕ∈H (N) = Hom(π (N),Z) is1
ﬁbered if and only if Ker(ϕ) is ﬁnitely generated.
7. Symplectic 4{manifolds with a free circle action
In this section we will prove Theorem 1.4.
7.1. Preliminaries. Westartbyrecallingsomeelementaryfactsaboutthealgebraic
topology of a 4–manifold M that carries a free circle action. The free circle action
renders M the total space of a circle bundle p: M → N over the orbit space, with
2Euler class e∈H (N). For the purpose of proving Theorem 1.4, we will see that we
can limit the discussion to the case where the Euler class is nontorsion, so we will
make this assumption for the rest of the subsection. The Gysin sequence reads
p p[e [e0 2 2 1 3H (N) H (N) H (M) H (N) H (N)
= = = = =
p\e \e
H (N) H (N) H (M) H (N) H (N),3 1 2 2 0
2 1where p : H (M) → H (N) denotes integration along the ﬁber. In particular we
have
p p2 20→⟨e⟩→H (N)−→H (M)−→ ker(·e)→ 0,
2where ⟨e⟩ is the cyclic subgroup of H (N) generated by the Euler class and where
1ker(·e)denotes the subgroup of elementsinH (N) whose pairing with the Euler class
vanishes. Ase is nontorsion, it follows thatb (M) = 2b (N)−2. It is not diﬃcult to2 1
+verify that sign(M) = 0, hence b (M) =b (N)−1.12
Letα: π (N)→G be a homomorphism to a ﬁnite group. We denote by π: N →1 G
N the regular G–cover of N. It is well known that b (N ) ≥ b (N). If π: M → N1 G 1
2is a circle bundle with Euler class e ∈ H (N) then α determines a regular G–cover
of M that we will denote (with slight abuse of notation) π: M →M. These coversG
are related by the commutative diagram
π
(1) M MG
π
N NG
2where the circle bundle p : M → N has Euler class e = π e ∈ H (N ), that isG G G G G
nontorsion as e is.10 STEFAN FRIEDL AND STEFANO VIDUSSI
7.2. Seiberg-Witten theory for symplectic manifolds with a circle action.
In this section we will apply, for the class of we are studying, Taubes’
nonvanishing theorem for Seiberg–Witten invariants of symplectic manifold to get a
restriction on the class of orbit spaces of a free circle action over a symplectic 4–
manifold. In order to do so, we need to understand the Seiberg-Witten invariants of
M. Again, we will limit the discussion here to the case where the Euler class e ∈
2H (N)isnottorsion. (Thetorsioncasewillbetreatedasacorollaryof[FV11a].) The
essential ingredient is the fact that the Seiberg-Witten invariants of M are related to
the Alexander polynomial ofN. Baldridge proved the following result, that combines
Corollaries 25 and 27 of [Ba03] (cf. also [Ba01]), to which we refer the reader for
deﬁnitions and results for Seiberg-Witten theory in this set-up:
Theorem 7.1. (Baldridge) Let M be a 4{manifold admitting a free circle action
2with nontorsion Euler classe∈H (N), whereN is the orbit space. Then the Seiberg-
2 2Witten invariant SW (κ) of a class κ = p ξ ∈ p H (N) ⊂ H (M) is given by theM
formula
∑
(2) SW (κ) = SW (ξ+le)∈Z,M N
l2Z
+in particular when b (M) = 1 it is independent on the chamber in which it was2
+calculated. Moreover, if b (M)> 1, these are the only basic classes.2
The Seiberg–Witten invariants ofN determine, via [MT96], the Alexander polyno-
mial of N. Assuming that a manifold M as above is symplectic, we will use Taubes’
constraints on its Seiberg-Witten invariants and Baldridge’s formula to get a con-
straint on the twisted Alexander polynomials ofN. We start with a technical lemma.
Lemma 7.2. Let (M,ω) be a symplectic manifold admitting a free circle action with
2nontorsion Euler class e ∈ H (N), where N is the orbit space. Then the canonical
2 2class K ∈ H (M) of the symplectic structure is the pull-back of a class ζ ∈ H (N),
where ζ is well{de ned up to the addition of a multiple of e.
+Proof. If b (M) > 1 this is a straightforward consequence of Theorem 7.1, as the2
canonical class by [Ta94] is a basic class ofM, hence must be the pull-back of a class
+2ofH (N). The case ofb (M) = 1 can be similarly obtained with a careful analysis of2
the chamber structure of the Seiberg-Witten invariants for classes that are not pull-
back, but it is possible to use a quicker argument. First, observe that, starting from
a closed curve inN representing a suitable element ofH (N), we can identify a torus1
T ⊂M ofself–intersectionzero,representingthegeneratorofacyclicsubgroupinthe
image of the map H (N) → H (M) in the homology Gysin sequence, that satisﬁes1 2
ω·[T]> 0. Second we can assume, by [Liu96], that K·ω≥ 0. (Otherwise M would
be a rational or ruled surface; using classical invariants, the only possibility would
2 2be M = S ×T , but then e = 0.) Also, as both signature and Euler characteristic
2of M vanish, K = 2χ(M) + 3σ(M) = 0. Omitting the case of K torsion, where
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