Results 1  10
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46
Discrete Morse Theory for Manifolds with Boundary
, 2012
"... We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain “Relative Morse Inequalities ” relating the homology of the manifold to the number of interior critical cells. We also ..."
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Cited by 14 (8 self)
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derive a Ball Theorem, in analogy to Forman’s Sphere Theorem. The main corollaries of our work are: (1) For each d ≥ 3andforeachk ≥ 0, there is a PL dsphere on which any discrete Morse function has more than k critical (d − 1)cells. (This solves a problem by Chari.) (2) For fixed d and k
Derived subdivisions make every PL sphere polytopal
 arXiv:1311.2965v2
"... Abstract. We give a simple proof that some iterated derived subdivision of every PL sphere is combinatorially equivalent to the boundary of a simplicial polytope, thereby resolving a problem of Billera (personal communication). I. Making any PL sphere polytopal. A subdivision of a simplicial comple ..."
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Cited by 1 (1 self)
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Abstract. We give a simple proof that some iterated derived subdivision of every PL sphere is combinatorially equivalent to the boundary of a simplicial polytope, thereby resolving a problem of Billera (personal communication). I. Making any PL sphere polytopal. A subdivision of a simplicial
Simplicial manifolds, bistellar flips and a 16vertex triangulation of the Poincaré homology 3sphere
 Math
, 2000
"... We present an algorithm based on bistellar operations that provides a useful tool for the construction of simplicial manifolds with few vertices. As an example, we obtain a 16vertex triangulation of the Poincaré homology 3sphere; we construct an infinite series of nonPL ddimensional spheres with ..."
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Cited by 47 (15 self)
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We present an algorithm based on bistellar operations that provides a useful tool for the construction of simplicial manifolds with few vertices. As an example, we obtain a 16vertex triangulation of the Poincaré homology 3sphere; we construct an infinite series of nonPL ddimensional spheres
CONSTRUCTING SIMPLICIAL BRANCHED COVERS
, 2007
"... Branched covers are applied frequently in topology most prominently in the construction of closed oriented PL dmanifolds. In particular, strong bounds for the number of sheets and the topology of the branching set are known for dimension d ≤ 4. On the other hand, Izmestiev and Joswig described how ..."
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closed oriented PL dmanifold is the partial unfolding of some polytopal dsphere.
Codimension two PL embeddings of spheres with nonstandard regular neighborhoods, arXiv:math.GT/0608653
 Chinese Ann. of Math., Ser. B
, 2006
"... Abstract. For a given polyhedron K ⊂ M the notation RM(K) denotes a regular neighborhood of K in M. We study the following problem: find all pairs (m, k) such that if K is a compact kpolyhedron and M a PL mmanifold, then RM(fK) ∼ = RM(gK), for each two homotopic PL embeddings f, g: K → M. We pro ..."
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Cited by 1 (0 self)
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prove that R S k+2(S k) ̸ ∼ = S k × D 2 for each k ≥ 2 and some PL sphere S k ⊂ S k+2 (even for any PL sphere S k ⊂ S k+2 having an isolated nonlocally flat point with the singularity S k−1 ⊂ S k+1 such that π1(S k+1 − S k−1) ̸ ∼ = Z). This paper is on the uniqueness of regular neighborhoods
Constructions of Cubical Polytopes
, 2004
"... In this thesis we consider cubical dpolytopes, convex bounded ddimensional polyhedra all of whose facets are combinatorially isomorphic to the (d − 1)dimensional standard cube. It is known that every cubical dpolytope P determines a PL immersion of an abstract closed cubical (d−2)manifold into t ..."
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Cited by 3 (1 self)
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)spheres, Babson and Chan have observed that every type of normal crossing PL immersion of a closed PL (d−2)manifold into an (d−1)sphere appears among the dual manifolds of some cubical PL (d − 1)sphere. No similar general result was available for cubical polytopes. The reason for this may be blamed
Classification of simplicial triangulations of topological manifolds
 Ann. of Math
"... In this note we announce theorems which classify simplicial (not necessarily combinatorial) triangulations of a given topological «manifold M, n> 7 (> 6 if dM = 0) , in terms of homotopy classes of lifts of the classifying map r: M —• BTOP for the stable topological tangent bundle of M to a c ..."
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Cited by 23 (0 self)
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classifying space BTRIn which we introduce below. The (homotopic) fiber of the natural map ƒ: BTRIn — • BTOP is described in terms of certain groups of PL homology 3spheres. We also give necessary and sufficient conditions for a closed topological «manifold M, n> 6, to possess a simplicial triangulation
Constructions of Cubical Polytopes vorgelegt von
"... In this thesis we consider cubical dpolytopes, convex bounded ddimensional polyhedra all of whose facets are combinatorially isomorphic to the (d − 1)dimensional standard cube. It is known that every cubical dpolytope P determines a PL immersion of an abstract closed cubical (d−2)manifold into t ..."
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)spheres, Babson and Chan have observed that every type of normal crossing PL immersion of a closed PL (d−2)manifold into an (d−1)sphere appears among the dual manifolds of some cubical PL (d − 1)sphere. No similar general result was available for cubical polytopes. The reason for this may be blamed
Metric Geometry and Collapsibility
, 2012
"... Cheeger’s finiteness theorem bounds the number of diffeomorphism types of manifolds with bounded curvature, diameter and volume; the Hadamard–Cartan theorem, as popularized by Gromov, shows the contractibility of all nonpositively curved simply connected metric length spaces. We establish a discret ..."
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simplex collapses simplicially, after d − 2 barycentric subdivisions. (This presents progress on an old question by Kirby and Lickorish.) (3) There are exponentially many geometric triangulations of Sd. (This interpolates between the result that polytopal dspheres are exponentially many, and the conjecture
The lower bound theorem and minimal triangulations of sphere bundles over the circle
, 2006
"... For integers d ≥ 2 and ε = 0 or 1, let S 1,d−1 (ε) denote the sphere product S 1 × S d−1 if ε = 0 and the twisted sphere product S 1 × − S d−1 if ε = 1. The main results of this paper are: (a) if d ≡ ε (mod 2) then S 1,d−1 (ε) has a unique minimal triangulation using 2d + 3 vertices, and (b) if d ≡ ..."
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Cited by 2 (1 self)
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For integers d ≥ 2 and ε = 0 or 1, let S 1,d−1 (ε) denote the sphere product S 1 × S d−1 if ε = 0 and the twisted sphere product S 1 × − S d−1 if ε = 1. The main results of this paper are: (a) if d ≡ ε (mod 2) then S 1,d−1 (ε) has a unique minimal triangulation using 2d + 3 vertices, and (b) if d
Results 1  10
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