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11
van Kampen’s embedding obstructions for discrete groups
 Invent. Math
, 2002
"... We give a lower bound to the dimension of a contractible manifold on which a given group can act properly discontinuously. In particular, we show that the nfold product of nonabelian free groups cannot act properly discontinuously on R 2n−1. 1 ..."
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Cited by 9 (2 self)
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We give a lower bound to the dimension of a contractible manifold on which a given group can act properly discontinuously. In particular, we show that the nfold product of nonabelian free groups cannot act properly discontinuously on R 2n−1. 1
Polytope Skeletons And Paths
 Handbook of Discrete and Computational Geometry (Second Edition ), chapter 20
"... INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent i ..."
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Cited by 6 (0 self)
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INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t
Higher minors and Van Kampen’s obstruction
 Math. Scand
"... We generalize the notion of graph minors to all (finite) simplicial complexes. For every two simplicial complexes H and K and every nonnegative integer m, we prove that if H is a minor of K then the non vanishing of Van Kampen’s obstruction in dimension m (a characteristic class indicating non embed ..."
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Cited by 6 (2 self)
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We generalize the notion of graph minors to all (finite) simplicial complexes. For every two simplicial complexes H and K and every nonnegative integer m, we prove that if H is a minor of K then the non vanishing of Van Kampen’s obstruction in dimension m (a characteristic class indicating non embeddability in the (m − 1)sphere) for H implies its non vanishing for K. As a corollary, based on results by Van Kampen [20] and Flores [5], if K has the dskeleton of the (2d+2)simplex as a minor, then K is not embeddable in the 2dsphere. We answer affirmatively a problem asked by Dey et. al. [3] concerning topologypreserving edge contractions, and conclude from it the validity of the generalized lower bound inequalities for a special class of triangulated spheres. 1
Dimension gaps between representability and collapsibility. Discrete Comput. Geom
 Comput. Geom
, 2008
"... A simplicial complex K is called drepresentable if it is the nerve of a collection of convex sets in R d; K is dcollapsible if it can be reduced to an empty complex by repeatedly removing a face of dimension at most d − 1 that is contained in a unique maximal face; and K is dLeray if every induce ..."
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Cited by 4 (4 self)
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A simplicial complex K is called drepresentable if it is the nerve of a collection of convex sets in R d; K is dcollapsible if it can be reduced to an empty complex by repeatedly removing a face of dimension at most d − 1 that is contained in a unique maximal face; and K is dLeray if every induced subcomplex of K has vanishing homology of dimension d and larger. It is known that drepresentable implies dcollapsible implies dLeray, and no two of these notions coincide for d ≥ 2. The famous Helly theorem and other important results in discrete geometry can be regarded as results about drepresentable complexes, and in many of these results “drepresentable ” in the assumption can be replaced by “dcollapsible ” or even “dLeray”. We investigate “dimension gaps ” among these notions, and we construct, for all d ≥ 1, a 2dLeray complex that is not (3d − 1)collapsible and a dcollapsible complex that is not (2d−2)representable. In the proofs we obtain two results of independent interest: (i) The nerve of every finite family of sets, each of size at most d, is dcollapsible. (ii) If the nerve of a simplicial complex K is drepresentable, then K embeds in R d. 1
Algebraic shifting and fvector theory
, 2007
"... This manuscript focusses on algebraic shifting and its applications to fvector theory of simplicial complexes and more general graded posets. It includes attempts to use algebraic shifting for solving the gconjecture for simplicial spheres, which is considered by many as the main open problem in f ..."
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Cited by 3 (1 self)
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This manuscript focusses on algebraic shifting and its applications to fvector theory of simplicial complexes and more general graded posets. It includes attempts to use algebraic shifting for solving the gconjecture for simplicial spheres, which is considered by many as the main open problem in fvector theory. While this goal has not been achieved, related results of independent interest were obtained, and are presented here. The operator algebraic shifting was introduced by Kalai over 20 years ago, with applications mainly in fvector theory. Since then, connections and applications of this operator to other areas of mathematics, like algebraic topology and combinatorics, have been found by different researchers. See Kalai’s recent survey [34]. We try to find (with partial success) relations between algebraic shifting and the following other areas: • Topological constructions on simplicial complexes. • Embeddability of simplicial complexes: into spheres and other manifolds.
Hardness of embedding simplicial complexes in R^d
, 2009
"... Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into d? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for a ..."
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Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into d? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for all k ≥ 3. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5sphere implies that EMBEDd→d and EMBED (d−1)→d are undecidable for each d ≥ 5. Our main result is NPhardness of EMBED2→4 and, more generally, of EMBEDk→d for all k, d with d ≥ 4 and d ≥ k ≥ (2d −2)/3. These dimensions fall outside the metastable range of a theorem of Haefliger and Weber, which characterizes embeddability using the deleted product obstruction. Our reductions are based on examples, due to Segal, Spie˙z, Freedman, Krushkal, Teichner, and Skopenkov, showing that outside the metastable range the deleted product obstruction is not sufficient to characterize embeddability.
ROBERTS ’ TYPE EMBEDDINGS AND CONVERSION OF THE TRANSVERSAL TVERBERG’S THEOREM
, 2005
"... Abstract. Here are two of our main results: Theorem 1. Let X be a normal space with dimX = n and m ≥ n + 1. Then the space C ∗ (X,R m) of all bounded maps from X into R m equipped with the uniform convergence topology contains a dense Gδsubset consisting of maps g such that g(X) ∩ Πd is at most (n ..."
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Abstract. Here are two of our main results: Theorem 1. Let X be a normal space with dimX = n and m ≥ n + 1. Then the space C ∗ (X,R m) of all bounded maps from X into R m equipped with the uniform convergence topology contains a dense Gδsubset consisting of maps g such that g(X) ∩ Πd is at most (n + d − m)dimensional for every ddimensional plane Πd in R m, where m − n ≤ d ≤ m. Theorem 2. Let X be a metrizable compactum with dimX ≤ n and m ≥ n + 1. Then, C(X,R m) contains a dense Gδsubset of maps g such that for any integers t, d, T with 0 ≤ t ≤ d ≤ m − n − 1 and d ≤ T ≤ m and any dplane Πd ⊂ R m parallel to some coordinate planes Πt ⊂ ΠT in R m, the inverse image g−1 (Πd) has at most q points, where q = d + 1 − t + n + (n + T − m)(d − t) n if n ≥ (m − n − T)(d − t) and q = 1 + m − n − d
Contents
, 2009
"... Abstract. We review a cochainfree treatment of the classical van Kampen obstruction ϑ to embeddability of an npolyhedron into R2n and consider several analogues and generalizations of ϑ, including an extraordinary lift of ϑ which in the manifold case has been studied by J.P. Dax. The following re ..."
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Abstract. We review a cochainfree treatment of the classical van Kampen obstruction ϑ to embeddability of an npolyhedron into R2n and consider several analogues and generalizations of ϑ, including an extraordinary lift of ϑ which in the manifold case has been studied by J.P. Dax. The following results are obtained. • The mod2 reduction of ϑ is incomplete, which answers a question of Sarkaria. • An odddimensional analogue of ϑ is a complete obstruction to linkless embeddability (=“intrinsic unlinking”) of the given npolyhedron in R2n+1. • A “blown up ” 1parameter version of ϑ is a universal type 1 invariant of singular knots, i.e. knots in R3 with a finite number of rigid transverse double points. We use it to decide in simple homological terms when a given integervalued type 1 invariant of singular knots admits an integral arrow diagram ( = Polyak–Viro) formula. • Settling a problem of Yashchenko in the metastable range, we obtain that every PL manifold N, nonembeddable in a given Rm, m ≥ 3(n+1), contains a subset X 2 such that no map N → Rm sends X and N \ X to disjoint sets. • We elaborate on McCrory’s analysis of the Zeeman spectral sequence to geometrically characterize “kcoconnected and locally kcoconnected ” polyhedra, which we embed in R2n−k for k < n−3 extending the Penrose–Whitehead–Zeeman theorem. 2