The mapping torus of a diffeomorphism

1984 ◽  
pp. 27-32
Author(s):  
Matthias Kreck
Keyword(s):  
Mapping Torus

The Nielsen-Thurston Classification

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Hyperbolic Geometry ◽  
Mapping Class Groups ◽  
Classification Theorem ◽  
Mapping Class ◽  
Fundamental Result ◽  
Class Groups ◽  
Mapping Torus ◽  
Higher Genus ◽  
Manifold Theory

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.

scholarly journals Omega vs. pi, and 6d anomaly cancellation

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Joe Davighi ◽  
Nakarin Lohitsiri
Keyword(s):  
Gauge Theories ◽  
Necessary Conditions ◽  
Homotopy Groups ◽  
Mapping Torus ◽  
Anomaly Cancellation ◽  
Global Anomaly ◽  
Gauge Anomalies ◽  
Local Anomalies ◽  
Dimensional Mapping

Abstract In this note we review the role of homotopy groups in determining non-perturbative (henceforth ‘global’) gauge anomalies, in light of recent progress understanding global anomalies using bordism. We explain why non-vanishing of πd(G) is neither a necessary nor a sufficient condition for there being a possible global anomaly in a d-dimensional chiral gauge theory with gauge group G. To showcase the failure of sufficiency, we revisit ‘global anomalies’ that have been previously studied in 6d gauge theories with G = SU(2), SU(3), or G2. Even though π6(G) ≠ 0, the bordism groups $$ {\Omega}_7^{\mathrm{Spin}}(BG) $$ Ω 7 Spin BG vanish in all three cases, implying there are no global anomalies. In the case of G = SU(2) we carefully scrutinize the role of homotopy, and explain why any 7-dimensional mapping torus must be trivial from the bordism perspective. In all these 6d examples, the conditions previously thought to be necessary for global anomaly cancellation are in fact necessary conditions for the local anomalies to vanish.

Homotopy equivalences and mapping torus projections

1980 ◽  
Vol 109 (1) ◽  
pp. 1-7 ◽  
Author(s):  
D. Coram ◽  
P. Duvall
Keyword(s):  
Mapping Torus

scholarly journals Ranks of mapping tori via the curve complex

2019 ◽  
Vol 2019 (748) ◽  
pp. 153-172 ◽  
Author(s):  
Ian Biringer ◽  
Juan Souto
Keyword(s):  
Fundamental Group ◽  
Orientable Surface ◽  
Curve Complex ◽  
Mapping Torus ◽  
Closed Orientable Surface ◽  
Mapping Tori

Abstract We show that if ϕ is a homeomorphism of a closed, orientable surface of genus g, and ϕ has large translation distance in the curve complex, then the fundamental group of the mapping torus {M_{\phi}} has rank {2g+1} .

scholarly journals Invariant incompressible surfaces in reducible 3-manifolds

2018 ◽  
Vol 39 (11) ◽  
pp. 3136-3143 ◽  
Author(s):  
CHRISTOFOROS NEOFYTIDIS ◽  
SHICHENG WANG
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Incompressible Surface ◽  
Mapping Torus ◽  
Incompressible Surfaces ◽  
Hyperbolic Automorphism

We study the effect of the mapping class group of a reducible 3-manifold $M$ on each incompressible surface that is invariant under a self-homeomorphism of $M$ . As an application of this study we answer a question of F. Rodriguez Hertz, M. Rodriguez Hertz, and R. Ures: a reducible 3-manifold admits an Anosov torus if and only if one of its prime summands is either the 3-torus, the mapping torus of $-\text{id}$ , or the mapping torus of a hyperbolic automorphism.

scholarly journals A remark on the η-invariant for automorphisms of hyperelliptic Riemann surfaces

2000 ◽  
Vol 62 (2) ◽  
pp. 177-182 ◽  
Author(s):  
Takayuki Morifuji
Keyword(s):  
Finite Order ◽  
Riemann Surfaces ◽  
Prime Order ◽  
Mapping Torus ◽  
Hyperelliptic Riemann Surfaces ◽  
Order Surface

We give a characterisation for the vanishing of the η-invariant of prime order automorphisms of hyperelliptic Riemann surfaces through the mapping torus construction. To this end, we introduce a notion of s-symmetry for finite order surface automorphisms.

scholarly journals Topological Entropy of Pseudo-Anosov Maps from a Typical Thurston’s Construction

2021 ◽  
Author(s):  
Hyungryul Baik ◽  
Inhyeok Choi ◽  
Dongryul M Kim
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Spectral Theorem ◽  
Mapping Torus ◽  
Moment Conditions ◽  
Salem Numbers ◽  
Teichmüller Metric ◽  
Hyperbolic Volume ◽  
Rank 2 ◽  
Extract Information

Abstract In this paper, we develop a way to extract information about a random walk associated with a typical Thurston’s construction. We first observe that a typical Thurston’s construction entails a free group of rank 2. We also present a proof of the spectral theorem for random walks associated with Thurston’s construction that have finite 2nd moment with respect to the Teichmüller metric. Its general case was remarked by Dahmani and Horbez. Finally, under a hypothesis not involving moment conditions, we prove that random walks eventually become pseudo-Anosov. As an application, we first discuss a random analogy of Kojima and McShane’s estimation of the hyperbolic volume of a mapping torus with pseudo-Anosov monodromy. As another application, we discuss non-probabilistic estimations of stretch factors from Thurston’s construction and the powers for Salem numbers to become the stretch factors of pseudo-Anosovs from Thurston’s construction.

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