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37
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 23 (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 12 (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.
REIDEMEISTER TORSION AND ANALYTIC TORSION OF DISCS
, 811
"... Abstract. We study the Reidemeister torsion and the analytic torsion of the m dimensional disc in the Euclidean m dimensional space, using the base for the homology defined by Ray and Singer in [20]. We prove that the Reidemeister torsion coincides with a power of the volume of the disc. We study th ..."
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Cited by 9 (4 self)
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Abstract. We study the Reidemeister torsion and the analytic torsion of the m dimensional disc in the Euclidean m dimensional space, using the base for the homology defined by Ray and Singer in [20]. We prove that the Reidemeister torsion coincides with a power of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the CheegerMüller theorem. We use a formula proved by Brüning and Ma [2], that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary [16]. Some of our results extend to the case of the cone over a sphere, in particular we evaluate directly the analytic torsion for a cone over the circle and over the two sphere. We compare the results obtained in the low dimensional cases. We also consider a different formula for the boundary term given by Dai and Fang [9], and we show that the result obtained using this formula is inconsistent with the direct calculation of the analytic torsion. 1.
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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Cited by 8 (2 self)
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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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Cited by 6 (0 self)
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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
Topology of moduli spaces of tropical curves with marked points
, 2008
"... In this paper we study topology of moduli spaces of tropical curves of genus g with n marked points. We view the moduli spaces as being imbedded in a larger space, which we call the moduli space of metric graphs with n marked points. We describe the shrinking bridges strong deformation retraction, ..."
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In this paper we study topology of moduli spaces of tropical curves of genus g with n marked points. We view the moduli spaces as being imbedded in a larger space, which we call the moduli space of metric graphs with n marked points. We describe the shrinking bridges strong deformation retraction, which leads to a substantial simplification of all these moduli spaces. In the rest of the paper, that reduction is used to analyze the case of genus 1. The corresponding moduli space is presented as a quotient space of a torus with respect to the conjugation Z2action; and furthermore, as a homotopy colimit over a simple diagram. The latter allows us to compute all Betti numbers of that moduli space with coefficients in Z2.
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.
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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Cited by 4 (0 self)
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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].