Mapping Class Groups of 3-Manifolds, Then and Now

Hong, Sungbok; McCullough, Darryl

doi:10.1090/conm/597/11768

Cited by 10 publications

(6 citation statements)

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…For arbitrary compact 3-manifolds, mapping class groups are only known to be finitely presented, see Conjecture F and the comments at the end of Section 4 of [42]. Notice that there is a subtle difference between the definition of our mapping class group and the mapping class group in [42], where they allow isotopies to move points on the boundary. But when the boundary is a union of spheres, there is a surjective map from our sphere-permuting mapping class group (Definition 2.1) to their mapping class group, whose kernel is a finite abelian group generated by sphere twists along boundaries.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

Aramayona¹,

Bux²,

Jonas³

et al. 2021

Preprint

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A Cantor manifold C is a non-compact manifold obtained by gluing (holed) copies of a fixed compact manifold Y in a tree-like manner. Generalizing classical families of groups due to Brin [10], Dehornoy [18] and 23,1], we introduce the asymptotic mapping class group B of C, whose elements are proper isotopy classes of selfdiffeomorphisms of C that are eventually trivial. The group B happens to be an extension of a Higman-Thompson group by a direct limit of mapping class groups of compact submanifolds of C.We construct an infinite-dimensional contractible cube complex X on which B acts. For certain well-studied families of manifolds, we prove that B is of type F8 and that X is CATp0q; more concretely, our methods apply for example when Y is diffeomorphic to S 1 ˆS1 , S 2 ˆS1 , or S n ˆSn for n ě 3. In these cases, B contains, respectively, the mapping class group of every compact surface with boundary; the automorphism group of the free group on k generators for all k; and an infinite family of (arithmetic) symplectic or orthogonal groups.In particular, if Y -S 2 or S 1 ˆS1 , our result gives a positive answer to [24, Problem 3] and [3, Question 5.32]. In addition, for Y -S 1 ˆS1 or S 2 ˆS1 , the homology of B coincides with the stable homology of the relevant mapping class groups, as studied by Harer [31] and .

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

Aramayona¹,

Bux²,

Jonas³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Then, the Dehn twist along the meridian of the boundary torus will be trivialized. By a result of McCullough (see for instance [61]), every Dehn twist along the longitude induces a diffeomorphism of the solid torus. Then, the complement of the unknot can carry an electric charge (by a Dehn twist) but no magnetic charge.…”

Section: Electric Charge As a Frame Of The Knot Complementmentioning

confidence: 99%

Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics

Asselmeyer-Maluga

2019

Symmetry

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In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston's geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement C(K) = S 3 \ (K × D 2 ) of a knot K carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which will be used in the description of fermions. Here, we choose the description of 3-manifolds by branched covers. Every 3-manifold can be described by a 3-fold branched cover of S 3 branched along a knot. In case of knot complements, one will obtain a 3-fold branched cover of the 3-disk D 3 branched along a 3-braid or 3-braids describing fermions. The whole approach will uncover new symmetries as induced by quantum and discrete groups. Using the Drinfeld-Turaev quantization, we will also construct a quantization so that quantum states correspond to knots. Particle properties like the electric charge must be expressed by topology, and we will obtain the right spectrum of possible values. Finally, we will get a connection to recent models of Furey, Stoica and Gresnigt using octonionic and quaternionic algebras with relations to 3-braids (Bilson-Thompson model).

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“…As in the proof of Theorem B, the invertibility of X implies that the V j are distinct smooth manifolds, relative to a fixed identification of ∂V j with N . But since N is a hyperbolic manifold, π 0 (Diff(N )) is finite [17,18] (a thorough discussion of such issues may be found in [23]). By Lemma 5.1, infinitely many of the V j are absolutely distinct.…”

Section: Explicit Examples Of Absolutely Exotic Corks and Anti-corksmentioning

confidence: 99%

Absolutely exotic compact 4-manifolds

Akbulut¹,

Ruberman

2016

Comment. Math. Helv.

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We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent to W that are absolutely exotic copies of each other. In this context, absolute means that the exotic structure is not relative to a particular parameterization of the boundary. Our examples are constructed by modifying a relatively exotic manifold by adding an invertible homology cobordism along its boundary. Applying this technique to corks (contractible manifolds with a diffeomorphism of the boundary that does not extend to a diffeomorphism of the interior) gives examples of absolutely exotic smooth structures on contractible 4-manifolds.

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Mapping Class Groups of 3-Manifolds, Then and Now

Cited by 10 publications

References 0 publications

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics

Absolutely exotic compact 4-manifolds

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