6,392 research outputs found
Links, two-handles, and four-manifolds
We show that only finitely many links in a closed 3-manifold share the same
complement, up to twists along discs and annuli. Using the same techniques, we
prove that by adding 2-handles on the same link we get only finitely many
smooth cobordisms between two given closed 3-manifolds.
As a consequence, there are finitely many smooth closed 4-manifolds
constructed from some Kirby diagram with bounded number of crossings, discs,
and strands, or from some Turaev special shadow with bounded number of
vertices. (These are the 4-dimensional analogues of Heegaard diagrams and
special spines for 3-manifolds.) We therefore get two filtrations on the set of
all smooth closed 4-manifolds with finite sets. The two filtrations are
equivalent after linear rescalings, and their cardinality grows at least as
n^{c*n}.Comment: 23 pages, 9 figures. Final versio
Hyperbolic three-manifolds that embed geodesically
We prove that every complete finite-volume hyperbolic 3-manifold that is
tessellated into right-angled regular polyhedra (dodecahedra or ideal
octahedra) embeds geodesically in a complete finite-volume connected orientable
hyperbolic 4-manifold , which is also tessellated into right-angled regular
polytopes (120-cells and ideal 24-cells). If is connected, then Vol() <
Vol(). This applies for instance to the Whitehead and the Borromean
links complements. As a consequence, the Borromean link complement bounds
geometrically a hyperbolic 4-manifold.Comment: 11 pages, 6 figures. Minor corrections from the previous version
Complexity of 3-manifolds
We give a summary of known results on Matveev's complexity of compact
3-manifolds. The only relevant new result is the classification of all closed
orientable irreducible 3-manifolds of complexity 10.Comment: 26 pages, 7 figures, minor correction
A finite set of local moves for Kirby calculus
We exhibit a finite set of local moves that connect any two surgery
presentations of the same 3-manifold via framed links in the three-sphere. The
moves are handle-slides and blow-downs/ups of a particular simple kind.Comment: 5 pages, 9 figures, minor changes following referee's suggestion
Countable groups are mapping class groups of hyperbolic 3-manifolds
We prove that for every countable group G there exists a hyperbolic
3-manifold M such that the isometry group of M, the mapping class group of M,
and the outer automorphism group of the fundamental group of M are isomorphic
to G.Comment: 15 pages, 6 figure
Dehn filling of the "magic" 3-manifold
We classify all the non-hyperbolic Dehn fillings of the complement of the
chain-link with 3 components, conjectured to be the smallest hyperbolic
3-manifold with 3 cusps. We deduce the classification of all non-hyperbolic
Dehn fillings of infinitely many 1-cusped and 2-cusped hyperbolic manifolds,
including most of those with smallest known volume. Among other consequences of
this classification, we mention the following:
- for every integer n we can prove that there are infinitely many hyperbolic
knots in the 3-sphere having exceptional surgeries n, n+1, n+2, n+3, with n+1,
n+2 giving small Seifert manifolds and n, n+3 giving toroidal manifolds;
- we exhibit a 2-cusped hyperbolic manifold that contains a pair of
inequivalent knots having homeomorphic complements;
- we exhibit a chiral 3-manifold containing a pair of inequivalent hyperbolic
knots with orientation-preservingly homeomorphic complements;
- we give explicit lower bounds for the maximal distance between small
Seifert fillings and any other kind of exceptional filling.Comment: 56 pages, 10 figures, 16 tables. Some consequences of the
classification adde
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