8,065 research outputs found
Remarks on the entropy of 3-manifolds
We give a simple combinatoric proof of an exponential upper bound on the
number of distinct 3-manifolds that can be constructed by successively
identifying nearest neighbour pairs of triangles in the boundary of a
simplicial 3-ball and show that all closed simplicial manifolds that can be
constructed in this manner are homeomorphic to . We discuss the problem of
proving that all 3-dimensional simplicial spheres can be obtained by this
construction and give an example of a simplicial 3-ball whose boundary
triangles can be identified pairwise such that no triangle is identified with
any of its neighbours and the resulting 3-dimensional simplicial complex is a
simply connected 3-manifold.Comment: 12 pages, 5 figures available from author
Infinite systolic groups are not torsion
We study -systolic complexes introduced by T. Januszkiewicz and J.
\'{S}wi\k{a}tkowski, which are simply connected simplicial complexes of
simplicial nonpositive curvature. Using techniques of filling diagrams we prove
that for the -skeleton of a -systolic complex is Gromov
hyperbolic. We give an elementary proof of the so-called Projection Lemma,
which implies contractibility of -systolic complexes. We also present a new
proof of the fact that an infinite group acting geometrically on a -systolic
complex is not torsion.Comment: Version 3, 27 pages, 10 figures. Major revision. Proof of Theorem 1.2
corrected, proof of Theorem 4.3 simplified, a reference to an alternative
proof of Theorem 7.4 added. Several definitions and lemmas adjusted and few
typos removed. Language and exposition improved. Version very similar to the
published versio
Simplicial Structure on Complexes
While chain complexes are equipped with a differential satisfying , their generalizations called -complexes have a differential
satisfying . In this paper we show that the lax nerve of the category
of chain complexes is pointwise adjoint equivalent to the d\'ecalage of the
simplicial category of -complexes. This reveals additional simplicial
structure on the lax nerve of the category of chain complexes which provides a
categorfication of the triangulated homotopy category of chain complexes. We
study this phenomena in general and present evidence that the axioms of
triangulated categories have simplicial origin
Systolic geometry and simplicial complexity for groups
Twenty years ago Gromov asked about how large is the set of isomorphism
classes of groups whose systolic area is bounded from above. This article
introduces a new combinatorial invariant for finitely presentable groups called
{\it simplicial complexity} that allows to obtain a quite satisfactory answer
to his question. Using this new complexity, we also derive new results on
systolic area for groups that specify its topological behaviour.Comment: 35 pages, 9 figure
Expoential bounds on the number of causal triangulations
We prove that the number of combinatorially distinct causal 3-dimensional
triangulations homeomorphic to the 3-dimensional sphere is bounded by an
exponential function of the number of tetrahedra. It is also proven that the
number of combinatorially distinct causal 4-dimensional triangulations
homeomorphic to the 4-sphere is bounded by an exponential function of the
number of 4-simplices provided the number of all combinatorially distinct
triangulations of the 3-sphere is bounded by an exponential function of the
number of tetrahedra.Comment: 30 pages, 9 figure
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