Alexander and Thurston norms of fibered 3-manifolds

Dunfield, Nathan M.

doi:10.2140/pjm.2001.200.43

Cited by 28 publications

(27 citation statements)

References 18 publications

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“…When the class φ represents a fibration, this bound becomes an equality, and ∆ N has peculiar properties, as shown from the following theorem, implicitly or explicitly proven, with different techniques, in [Du01], [McM02], [Vi03], and [FK06]. Remember that Thurston [Th86] showed that if a class φ ∈ H 1 (N ) is fibered then it lies in the cone over some topdimensional face F T .…”

Section: Symplectic Products and The Ordinary Alexander Polynomialmentioning

confidence: 99%

Twisted Alexander Polynomials and Symplectic Structures

Friedl¹,

Vidussi²

2008

ajm

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Abstract. Let N be a closed, oriented 3-manifold. A folklore conjecture states that S 1 × N admits a symplectic structure only if N admits a fibration over the circle. The purpose of this paper is to provide evidence to this conjecture studying suitable twisted Alexander polynomials of N , and showing that their behavior is the same as of those of fibered 3-manifolds. In particular, we will obtain new obstructions to the existence of symplectic structures and to the existence of symplectic forms representing certain cohomology classes of S 1 × N . As an application of these results we will show that S 1 × N (P ) does not admit a symplectic structure, where N (P ) is the 0-surgery along the pretzel knot P = (5, −3, 5), answering a question of Peter Kronheimer.

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Section: Symplectic Products and The Ordinary Alexander Polynomialmentioning

confidence: 99%

Twisted Alexander Polynomials and Symplectic Structures

Friedl¹,

Vidussi²

2008

ajm

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“…In many cases it is easier to find representations ofM . This approach allows us to determine the Thurston norm ball of Dunfield's link [Du01] (see Section 5.2).…”

Section: Theorem 32 (Main Theorem 2) Let M Be a 3-manifold With B 1mentioning

confidence: 99%

Twisted Alexander norms give lower bounds on the Thurston norm

Friedl

Kim

2008

Trans. Amer. Math. Soc.

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Abstract. We introduce twisted Alexander norms of a compact connected orientable 3-manifold with first Betti number greater than one, generalizing norms of McMullen and Turaev. We show that twisted Alexander norms give lower bounds on the Thurston norm of a 3-manifold. Using these we completely determine the Thurston norm of many 3-manifolds which are not determined by norms of McMullen and Turaev.

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“…Theorem 5.7 easily gives A simple example is shown in Figure 13. The BNS picture can also be connected to the Alexander polynomial, in particular to the coefficients which occur at the vertices of the Newton polygon [9].…”

Section: Propositionmentioning

confidence: 99%

A random tunnel number one 3–manifold does not fiber over the circle

Dunfield

Thurston

2006

Geom. Topol.

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We address the question: how common is it for a 3-manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3-manifolds with tunnel number one, we provide compelling theoretical and experimental evidence that fibering is a very rare property. Indeed, in various precise senses it happens with probability 0. Our main theorem is that this is true for a measured lamination model of random tunnel number one 3-manifolds.The first ingredient is an algorithm of K Brown which can decide if a given tunnel number one 3-manifold fibers over the circle. Following the lead of Agol, Hass and W Thurston, we implement Brown's algorithm very efficiently by working in the context of train tracks/interval exchanges. To analyze the resulting algorithm, we generalize work of Kerckhoff to understand the dynamics of splitting sequences of complete genus 2 interval exchanges. Combining all of this with a "magic splitting sequence" and work of Mirzakhani proves the main theorem.The 3-manifold situation contrasts markedly with random 2-generator 1-relator groups; in particular, we show that such groups "fiber" with probability strictly between 0 and 1. 57R22; 57N10, 20F05We dedicate this paper to the memory of Raoul Bott (1923Bott ( -2005, a wise teacher and warm friend, always searching for the simplicity at the heart of mathematics.

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Alexander and Thurston norms of fibered 3-manifolds

Cited by 28 publications

References 18 publications

Twisted Alexander Polynomials and Symplectic Structures

Twisted Alexander Polynomials and Symplectic Structures

Twisted Alexander norms give lower bounds on the Thurston norm

A random tunnel number one 3–manifold does not fiber over the circle

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