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• Publié par: zieg
• Nombre de lectures: 11
• Langue: English

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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
Pro