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24
Homology algorithm based on acyclic subspace
"... We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace ma ..."
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Cited by 17 (9 self)
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We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace may be performed in linear time. This significantly reduces the amount of data that needs to be processed in the algebraic way, and in practice it proves itself to be significantly more efficient than other available cubical homology algorithms.
Simple homotopy types of HomComplexes, neighbourhood complexes, Lovász complexes, and atom crosscut complexes
, 2008
"... In this paper we provide concrete combinatorial formal deformation algorithms, namely sequences of elementary collapses and expansions, which relate various previously extensively studied families of combinatorially defined polyhedral complexes. To start with, we give a sequence of elementary coll ..."
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Cited by 10 (3 self)
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In this paper we provide concrete combinatorial formal deformation algorithms, namely sequences of elementary collapses and expansions, which relate various previously extensively studied families of combinatorially defined polyhedral complexes. To start with, we give a sequence of elementary collapses leading from the barycentric subdivision of the neighborhood complex to the Lovász complex of a graph. Then, for an arbitrary lattice L we describe a formal deformation of the barycentric subdivision of the atom crosscut complex Γ(L) to its order complex ∆ ( ¯ L). We proceed by proving that the complex of sets bounded from below J (L) can also be collapsed to ∆ ( ¯ L). Finally, as a pinnacle of our project, we apply all these results to certain graph complexes. Namely, by describing an explicit formal deformation, we prove that, for any graph G, the neighborhood complex N(G) and the polyhedral complex Hom (K2, G) have the same simple homotopy type in the sense of Whitehead.
On Novikovtype Conjectures
 Zbl 0992.58012 MR 1847591
, 2001
"... We also added a brief epilogue, essentially “What there wasn’t time for. ” Although the focus of the conference was on noncommutative geometry, the topic discussed was conventional commutative motivations for the circle of ideas related to the Novikov and BaumConnes conjectures. While the article i ..."
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We also added a brief epilogue, essentially “What there wasn’t time for. ” Although the focus of the conference was on noncommutative geometry, the topic discussed was conventional commutative motivations for the circle of ideas related to the Novikov and BaumConnes conjectures. While the article is mainly expository, we present here a few new results (due to the two of us). It is interesting to note that while the period from 80’s through the mid90’s has shown a remarkable convergence between index theory and surgery theory (or more generally, the classification of manifolds) largely motivated by the Novikov conjecture, most recently, a number of divergences has arisen. Possibly, these subjects are now diverging, but it also seems plausible that we are only now close to discovering truly deep phenomena and that the difference between these subjects is just one of these. Our belief is that, even after decades of mining this vein, the gold is not yet all gone. As the reader might guess from the title, the focus of these notes is not quite on the Novikov conjecture itself, but rather on a collection of problems that are suggested by heuristics, analogies and careful consideration of consequences. Many of the related conjectures are false, or, as far as we know, not directly mathematically related to the original conjecture; this is a good thing: we learn about the subtleties of the original problem, the boundaries of the associated phenomenon, and get to learn about other realms of mathematics.
COLLAPSING ALONG MONOTONE POSET MAPS
, 2008
"... Abstract. We introduce the notion of nonevasive reduction, and show that for any monotone poset map ϕ: P → P, the simplicial complex ∆(P) NEreduces to ∆(Q), for any Q ⊇ Fix ϕ. As a corollary, we prove that for any orderpreserving map ϕ: P → P satisfying ϕ(x) ≥ x, for any x ∈ P, the simplicial com ..."
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Abstract. We introduce the notion of nonevasive reduction, and show that for any monotone poset map ϕ: P → P, the simplicial complex ∆(P) NEreduces to ∆(Q), for any Q ⊇ Fix ϕ. As a corollary, we prove that for any orderpreserving map ϕ: P → P satisfying ϕ(x) ≥ x, for any x ∈ P, the simplicial complex ∆(P) collapses to ∆(ϕ(P)). We also obtain a generalization of Crapo’s closure theorem. 1. Order complexes, collapsing and NEreduction. For a poset P we let ∆(P) denote its nerve: the simplicial complex whose simplices are all chains of P. For a simplicial complex X we let V (X) denote the set of its vertices. An elementary collapse in a simplicial complex X is a removal of two open simplices σ and τ from X, such that dimσ = dim τ + 1, and σ is the only simplex of X, different from τ itself, which contains the simplex τ in its closure. When Y is a subcomplex of X, we say that X collapses onto Y if there exists a sequence of elementary collapses leading from X to Y; in this case we write X ց Y (or, equivalently, Y ր X). Definition 1.1. (1) A finite nonempty simplicial complex X is called nonevasive if either X is a point, or, inductively, there exists a vertex v of X, such that both X \ {v} and lkXv are nonevasive. (2) For two nonempty simplicial complexes X and Y we write X ցNE Y (or, equivalently, Y րNE X), if there exists a sequence X = A1 ⊃ A2 ⊃ · · · ⊃ At = Y, such that for all i ∈ {1,...,t − 1} we can write V (Ai) = V (Ai+1) ∪ {xi}, so that lkAi xi is nonevasive. We recall, that the notion of nonevasive simplicial complexes was introduced in [KSS], and was initially motivated by the complexitytheoretic considerations. For further connections to topology and more facts on nonevasiveness we refer to [Ku90, We99]. Recently an interesting connection has
Simple homotopy type of some combinatorially defined complexes
, 2005
"... Abstract. In this paper we provide concrete combinatorial formal deformation algorithms, namely sequences of elementary collapses and expansions, which relate various previously extensively studied families of combinatorially defined polyhedral complexes. To start with, we give a sequence of element ..."
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Cited by 4 (2 self)
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Abstract. In this paper we provide concrete combinatorial formal deformation algorithms, namely sequences of elementary collapses and expansions, which relate various previously extensively studied families of combinatorially defined polyhedral complexes. To start with, we give a sequence of elementary collapses leading from the barycentric subdivision of the neighborhood complex to the Lovász complex of a graph. Then, for an arbitrary lattice L we describe a formal deformation of the barycentric subdivision of the atom crosscut complex Γ(L) to its order complex ∆ ( ¯ L). We proceed by proving that the complex of sets bounded from below J (L) can also be collapsed to ∆ ( ¯ L). Finally, as a pinnacle of our project, we apply all these results to certain graph complexes. Namely, by describing an explicit formal deformation, we prove that, for any graph G, the neighborhood complex N(G) and the polyhedral complex Hom (K2, G) have the same simple homotopy type in the sense of Whitehead.
Waldhausen's Nil Groups and Continuously Controlled KTheory
, 1997
"... Let \Gamma = \Gamma 1 G \Gamma 2 be the pushout of two groups \Gamma i , i = 1; 2; over a common subgroup G, and H be the double mapping cylinder of the corresponding diagram of classifying spaces B \Gamma 1 / BG ! B \Gamma 2 . Denote by ¸ the diagram I p / H 1 ! X = H , where p is the natural ..."
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Cited by 3 (2 self)
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Let \Gamma = \Gamma 1 G \Gamma 2 be the pushout of two groups \Gamma i , i = 1; 2; over a common subgroup G, and H be the double mapping cylinder of the corresponding diagram of classifying spaces B \Gamma 1 / BG ! B \Gamma 2 . Denote by ¸ the diagram I p / H 1 ! X = H , where p is the natural map onto the unit interval. We show that the f Nil groups which occur in Waldhausen's description of K (Z\Gamma) coincide with the continuously controlled groups e K cc (¸), defined by Anderson and Munkholm. This also allows us to identify the continuously controlled groups e K cc (¸ + ) which are known to form a homology theory in the variable ¸, with the "homology part" in Waldhausen's description of K \Gamma1 (Z\Gamma). A similar result is also obtained for HNN extensions.
On the top homology of hypergraph matching complexes
 in preparation. FIBER THEOREMS 23
"... Abstract. We investigate the representation of a symmetric group Sn on the homology of its Quillen complex at a prime p. For homology groups in small codimension, we derive an explicit formula for this representation in terms of the representations of symmetric groups on homology groups of puniform ..."
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Abstract. We investigate the representation of a symmetric group Sn on the homology of its Quillen complex at a prime p. For homology groups in small codimension, we derive an explicit formula for this representation in terms of the representations of symmetric groups on homology groups of puniform hypergraph matching complexes. We conjecture an explicit formula for the representation of Sn on the top homology group of the corresponding hypergraph matching complex when n ≡ 1 mod p. Our conjecture follows from work of Bouc when p = 2, and we prove the conjecture when p = 3. 1.
Simplicial simplehomotopy of flag complexes of graphs
, 2009
"... A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1skeleton taken as a graph. In this article, by introducing the notion of sdismantlability, we shall define the shomotopy type of a graph and show in particular that two finite graphs hav ..."
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A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1skeleton taken as a graph. In this article, by introducing the notion of sdismantlability, we shall define the shomotopy type of a graph and show in particular that two finite graphs have the same shomotopy type if, and only if, the two flag complexes determined by these graphs have the same simplicial simplehomotopy type (Theorem 2.10, part 1). This result is closely related to similar results established by Barmak and Minian ([2]) in the framework of posets and we give the relation between the two approaches (theorems 3.5 and 3.7). We conclude with a question about the relation between the shomotopy and the graph homotopy defined in [5].