By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton
The sector of 3-manifold topology has made nice strides ahead in view that 1982 while Thurston articulated his influential record of questions. basic between those is Perelman's evidence of the Geometrization Conjecture, yet different highlights comprise the Tameness Theorem of Agol and Calegari-Gabai, the outside Subgroup Theorem of Kahn-Markovic, the paintings of clever and others on unique dice complexes, and, ultimately, Agol's evidence of the digital Haken Conjecture. This publication summarizes a majority of these advancements and offers an exhaustive account of the present cutting-edge of 3-manifold topology, particularly targeting the implications for primary teams of 3-manifolds. because the first e-book on 3-manifold topology that includes the intriguing growth of the final twenty years, will probably be a useful source for researchers within the box who desire a reference for those advancements. It additionally offers a fast moving advent to this fabric. even supposing a few familiarity with the elemental workforce is usually recommended, little different earlier wisdom is believed, and the booklet is out there to graduate scholars. The booklet closes with an intensive checklist of open questions with a view to even be of curiosity to graduate scholars and tested researchers. A booklet of the eu Mathematical Society (EMS). dispensed in the Americas through the yankee Mathematical Society.
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Extra info for 3-Manifold Groups
One could formulate the theorem more succinctly: if g = 1 and Cπ (g) is non-cyclic, then there exists a component C of the characteristic submanifold of N and h ∈ π such that g ∈ h π1 (C) h−1 and Cπ (g) = hCπ1 (C) (h−1 gh)h−1 . 1 which makes use of the deep results of Jaco–Shalen and Johannson and of the Geometrization Theorem for nonHaken manifolds. Alternatively the theorem can be proved using the Geometrization Theorem much more explicitly—we refer to [Fri11] for details. Proof. We first consider the case that N is hyperbolic.
7) If N is a Seifert fibered manifold, then N admits a fixed-point free S1 action, and Diff(N) thus contains torsion elements of arbitrarily large order. 1] showed that if N is a closed irreducible 3-manifold which is not Seifert fibered, then there is a bound on the order of finite subgroups of Diff(N). For 3-manifolds which are spherical or not prime the map Φ is in general neither injective nor surjective. See [Gab94b, McC90, McC95] for more information. Now we give a few more situations in which topological information can be ‘directly’ obtained from the fundamental group.
Furthermore we denote by Diff0 (N) the identity component of Diff(N). The quotient Diff(N)/Diff0 (N) is denoted by M (N). , the quotient of the group of automorphisms of π by its normal subgroup of inner automorphisms of π). If N = S1 ×D2 is a compact, irreducible 3-manifold which is not an I-bundle over a surface, then the natural morphism Φ : M (N) → ϕ ∈ Out(π) : ϕ preserves the peripheral structure is injective. 2]), Boileau–Otal [BO86], [BO91, Th´eor`eme 3], and the Geometrization Theorem. (1) A Seifert fibered manifold is called small if it is not Haken.
3-Manifold Groups by Matthias Aschenbrenner, Stefan Friedl, Henry Wilton
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