Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0407392

A set of moves for Johansson representation of 3-manifolds. An outline

Rubén Vigara
View original source ↗Read source document ↗Export metadata ↓

Abstract

A Dehn sphere in a closed 3-manifold M is a 2-sphere immersed in M with only double curve and triple point singularities. The Dehn sphere S fills M if it defines a cell-decomposition of M. The inverse image in S^{2} of the double curves of S is the Johansson diagram of S and if S fills M it is possible to reconstruct M from the diagram. A Johansson representation of M is the Johansson diagram of a filling Dehn sphere of M. In a recent paper of J. M. Montesinos it is proved that every closed 3-manifold has a Johansson representation coming from a nulhomotopic filling Dehn sphere. In this paper a set of moves for Johansson representations of 3-manifolds is given. In a forthcoming paper it is proved that this set of moves suffices for relating different Johansson representations of the same 3-manifold coming from nulhomotopic filling Dehn spheres. The proof of this result is outlined here.

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Rubén Vigara. 2004-07-27. A set of moves for Johansson representation of 3-manifolds. An outline. https://arxiv.org/abs/math/0407392

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Relativization of symmetries on quandles

This paper introduces relative versions of the inner automorphism group and the transvection group associated with surjective quandle homomorphisms. By using the relative inner automorphism group, we define a notion of connectedness for surjective homomorphisms. We characterize connected homomorphisms algebraically as quotient maps, and use the relative transvection group to establish a maximal connected--covering factorization for arbitrary surjections. We also introduce strongly connected homomorphisms via the relative transvection group, and classify them in terms of perfect normal subgroups of inner automorphism groups. A later part of this paper studies surjective homomorphisms for which the relative inner automorphism group acts $2$-transitively on each fiber. Under this assumption, we classify the possible quandle structures of the finite fibers.

math.GT↗
arXiv2026

Mapping class groups of simply connected closed spin 5-manifolds with no 2- and 3-torsion in homology

We show that the Torelli group of a simply connected closed 5-manifold $M$, which is spin and has no 2-torsion elements in homology, is isomorphic to the bordism group $Ω^{\mathrm{Spin}}_6(K(H_2(M), 2))$. For $H_2(M)$ with no 2- and 3-torsion we determine this bordism group, and give explicit constructions for the generators of the Torelli group. Furthermore, for the $g$-fold connected sum ${\#}^g(S^2 \times S^3)$ we completely determine its mapping class group. We apply our results to compute the stabilization and abelianization of the mapping class group of ${\#}^g(S^2 \times S^3)$, determine the group of isotopy classes of diffeomorphisms of $M$ that are homotopic to the identity, and study the embeddings of $S^3$ in $S^2 \times S^3$.

math.GT↗
arXiv2026

Conformal tetrahedra and global rigidity of sphere packings in constant curvature background geometry

In this paper, we give the dual characterization of conformal tetrahedra in constant curvature spaces, which develops a geometric technique to obtain the global rigidity of (ideal) hyperbolic, Euclidean, spherical sphere packings on a tetrahedron. Moreover, we introduce ideal hyperbolic sphere packings and extend the variational principle to obtain the global rigidity of (ideal) hyperbolic sphere packings on 3-manifolds. We also provide a characterization of conformal $n(\ge3)$-simplices in constant curvature spaces, which is useful for studying higher-dimensional sphere packings.

math.GT↗