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Register a new userThe triangulation complexity of fibred 3-manifolds
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The triangulation complexity of a closed orientable 3-manifold is the minimal number of tetrahedra in any triangulation of the manifold. The main theorem of the paper gives upper and lower bounds on the triangulation complexity of any closed orientable hyperbolic 3-manifold that fibres over the circle. We show that the triangulation complexity of the manifold is equal to the translation length of the monodromy action on the mapping class group of the fibre, up to a bounded factor, where the bound depends only on the genus of the fibre.
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We show the problem of counting homomorphisms from the fundamental group of a homology $3$-sphere $M$ to a finite, non-abelian simple group $G$ is #P-complete, in the case that $G$ is fixed and $M$ is the computational input. Similarly, deciding if there is a non-trivial homomorphism is NP-complete. In both reductions, we can guarantee that every non-trivial homomorphism is a surjection. As a corollary, for any fixed integer $m ge 5$, it is NP-complete to decide whether $M$ admits a connected $m$-sheeted covering. Our construction is inspired by universality results in topological quantum computation. Given a classical reversible circuit $C$, we construct $M$ so that evaluations of $C$ with certain initialization and finalization conditions correspond to homomorphisms $pi_1(M) to G$. An intermediate state of $C$ likewise corresponds to a homomorphism $pi_1(Sigma_g) to G$, where $Sigma_g$ is a pointed Heegaard surface of $M$ of genus $g$. We analyze the action on these homomorphisms by the pointed mapping class group $text{MCG}_*(Sigma_g)$ and its Torelli subgroup $text{Tor}_*(Sigma_g)$. By results of Dunfield-Thurston, the action of $text{MCG}_*(Sigma_g)$ is as large as possible when $g$ is sufficiently large; we can pass to the Torelli group using the congruence subgroup property of $text{Sp}(2g,mathbb{Z})$. Our results can be interpreted as a sharp classical universality property of an associated combinatorial $(2+1)$-dimensional TQFT.
We present an algorithm for producing Delaunay triangulations of manifolds. The algorithm can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. Given a set of sample points and an atlas on a compact manifold, a manifold Delaunay complex is produced provided the transition functions are bi-Lipschitz with a constant close to 1, and the sample points meet a local density requirement; no smoothness assumptions are required. If the transition functions are smooth, the output is a triangulation of the manifold. The output complex is naturally endowed with a piecewise flat metric which, when the original manifold is Riemannian, is a close approximation of the original Riemannian metric. In this case the ouput complex is also a Delaunay triangulation of its vertices with respect to this piecewise flat metric.
We classify those compact 3-manifolds with incompressible toral boundary whose fundamental groups are residually free. For example, if such a manifold $M$ is prime and orientable and the fundamental group of $M$ is non-trivial then $M cong Sigmatimes S^1$, where $Sigma$ is a surface.
In this paper we explore the topological properties of self-replicating, 3-dimensional manifolds, which are modeled by idempotents in the (2+1)-cobordism category. We give a classification theorem for all such idempotents. Additionally, we characterize biologically interesting ways in which self-replicating 3-manifolds can embed in $mathbb{R}^3$.
We prove that cubical simplicial volume of oriented closed 3-manifolds is equal to one fifth of ordinary simplicial volume.
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