By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton
The sector of 3-manifold topology has made nice strides ahead on the grounds that 1982 whilst Thurston articulated his influential record of questions. fundamental between those is Perelman's evidence of the Geometrization Conjecture, yet different highlights contain the Tameness Theorem of Agol and Calegari-Gabai, the outside Subgroup Theorem of Kahn-Markovic, the paintings of clever and others on targeted dice complexes, and, ultimately, Agol's evidence of the digital Haken Conjecture. This publication summarizes most of these advancements and offers an exhaustive account of the present state-of-the-art of 3-manifold topology, specially concentrating on the results for primary teams of 3-manifolds. because the first booklet on 3-manifold topology that comes with the intriguing development of the final 20 years, will probably be a useful source for researchers within the box who desire a reference for those advancements. It additionally offers a fast paced advent to this fabric. even though a few familiarity with the elemental team is usually recommended, little different prior wisdom is believed, and the ebook is on the market to graduate scholars. The booklet closes with an intensive record of open questions for you to even be of curiosity to graduate scholars and confirmed researchers. A e-book of the ecu Mathematical Society (EMS). allotted in the Americas via the yank Mathematical Society.
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Clearly π = Cπ (G). 1 that either N is Seifert fibered, π = Z, or π = Z2 . But the latter case also implies that N is one of S1 × D2 , T 2 × I, or K 2 × I. In particular, N is again Seifert fibered. Remarks. (1) This theorem was proved before the Geometrization Theorem: (a) Casson–Jungreis [CJ94] and Gabai [Gab92], extending earlier work of Tukia [Tuk88a, Tuk88b], showed that every word-hyperbolic group with boundary that is homeomorphic to S1 acts properly discontinuously and cocompactly on H2 with finite kernel.
Tm the JSJ-tori of N. We assume that they are ordered such that the tori T1 , . . , Tn do not bound copies of K 2 × I and that for i = n + 1, . . , m, each Ti cobounds a copy of K 2 × I. Then the geometric decomposition surface of N is given by T1 ∪ · · · ∪ Tn ∪ Kn+1 ∪ · · · ∪ Km . Conversely, if T1 ∪ · · · ∪ Tn ∪ Kn+1 ∪ · · · ∪ Km is the geometric decomposition such that T1 , . . , Tn are tori and Kn+1 , . . 9 The Geometric Decomposition Theorem 21 JSJ-tori are given by T1 ∪ · · · ∪ Tn ∪ ∂ νKn+1 ∪ · · · ∪ ∂ νKm .
1] for details and for [Hat82, HO89, Rat90, LiR91, SZ01, ChT07, HoSh07, Gar11, DG12] for extensions of this result. 9. (4) The Thurston norm H 1 (N; R) → R≥0 measures the minimal complexity of surfaces dual to cohomology classes. 3 for a precise definition and for details. (a) It follows from [FeG73, Theorem 1] that if N is a closed 3-manifold with b1 (N) = 1, then the Thurston norm can be recovered in terms of splittings of fundamental groups along surface groups. 3] this also gives a group-theoretic way to recover the genus of a knot in S3 .