• Title/Summary/Keyword: Euler Characteristic

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CRITICAL VIRTUAL MANIFOLDS AND PERVERSE SHEAVES

  • Kiem, Young-Hoon;Li, Jun
    • Journal of the Korean Mathematical Society
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    • v.55 no.3
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    • pp.623-669
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    • 2018
  • In Donaldson-Thomas theory, moduli spaces are locally the critical locus of a holomorphic function defined on a complex manifold. In this paper, we develop a theory of critical virtual manifolds which are the gluing of critical loci of holomorphic functions. We show that a critical virtual manifold X admits a natural semi-perfect obstruction theory and a virtual fundamental class $[X]^{vir}$ whose degree $DT(X)=deg[X]^{vir}$ is the Euler characteristic ${\chi}_{\nu}$(X) weighted by the Behrend function ${\nu}$. We prove that when the critical virtual manifold is orientable, the local perverse sheaves of vanishing cycles glue to a perverse sheaf P whose hypercohomology has Euler characteristic equal to the Donaldson-Thomas type invariant DT(X). In the companion paper, we proved that a moduli space X of simple sheaves on a Calabi-Yau 3-fold Y is a critical virtual manifold whose perverse sheaf categorifies the Donaldson-Thomas invariant of Y and also gives us a mathematical theory of Gopakumar-Vafa invariants.

Cross-index of a Graph

  • Kawauchi, Akio;Shimizu, Ayaka;Yaguchi, Yoshiro
    • Kyungpook Mathematical Journal
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    • v.59 no.4
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    • pp.797-820
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    • 2019
  • For every tree T, we introduce a topological invariant, called the T-cross-index, for connected graphs. The T-cross-index of a graph is a non-negative integer or infinity according to whether T is a tree basis of the graph or not. It is shown how this cross-index is independent of the other topological invariants of connected graphs, such as the Euler characteristic, the crossing number and the genus.

SIZE DISTRIBUTION OF ONE CONNECTED COMPONENT OF ELLIPTIC RANDOM FIELD

  • Alodat, M.T.
    • Journal of the Korean Statistical Society
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    • v.36 no.4
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    • pp.479-488
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    • 2007
  • The elliptic random field is an extension to the Gaussian random field. We proved a theorem which characterizes the elliptic random field. We proposed a heuristic approach to derive an approximation to the distribution of the size of one connected component of its excursion set above a high threshold. We used this approximation to approximate the distribution of the largest cluster size. We used simulation to compare the approximation with the exact distribution.

SOME MANIFOLDS WITH NONZERO EULER CHARACTERISTIC AS PL FIBRATORS

  • Im, Young-Ho
    • Honam Mathematical Journal
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    • v.29 no.3
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    • pp.327-339
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    • 2007
  • Approximate fibrations form a useful class of maps. By definition fibrators provide instant detection of maps in this class, and PL fibrators do the same in the PL category. We show that every closed s-hopfian t-aspherical manifold N with some algebraic conditions and X(N) $\neq$ 0 is a codimension-(2t + 2) PL fibrator.

COMPUTATION OF NIELSEN NUMBERS FOR CERTAIN MAPS OF HYPERBOLIC SURFACES

  • Kim, Seung Won
    • Journal of the Chungcheong Mathematical Society
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    • v.28 no.2
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    • pp.243-249
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    • 2015
  • Let X be a closed surface for which the Euler characteristic $_{\mathcal{X}}(X)$ is negative, and let $f:X{\rightarrow}X$ be a self-map that is not surjective. In this short paper, we prove that we can compute the Nielsen number of f, N(f), under some algebraic conditions.

PARTIALLY ASHPHERICAL MANIFOLDS WITH NONZERO EULER CHARACTERISTIC AS PL FIBRATORS

  • Im, Young-Ho;Kim, Yong-Kuk
    • Journal of the Korean Mathematical Society
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    • v.43 no.1
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    • pp.99-109
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    • 2006
  • Approximate fibrations form a useful class of maps. By definition fibrators provide instant detection of maps in this class, and PL fibrators do the same in the PL category. We show that every closed s-hopfian t-aspherical manifold N with sparsely Abelian, hopfian fundamental group and X(N) $\neq$ 0 is a codimension-(t + 1) PL fibrator.

ON THE ACTIONS OF HIGMAN-THOMPSON GROUPS BY HOMEOMORPHISMS

  • Kim, Jin Hong
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.449-457
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    • 2020
  • The aim of this short paper is to show some rigidity results for the actions of certain finitely presented groups by homeomorphisms. As an interesting and special case, we show that the actions of Higman-Thompson groups by homeomorphisms on a cohomology manifold with a non-zero Euler characteristic should be trivial. This is related to the wellknown Zimmer program and shows that the actions by homeomorphism could be very much different from those by diffeomorphisms.